Solve for x: Solution. The LCD of all the denominators is 10x. Multiply both sides of the D equation by 10x and solve the resolving equation. Factor each denominator in the rational expression. Multiply the LCD to both sides of the equation to remove the denominators. PY Upon reaching this step, we can use strategies for solving polynomial equations.
O or C or Since makes the original equat ion undefined, is the only solution. The team has already won 12 out of their 25 games. The equation is. Jens walks 5 kilometers from his house to Quiapo to buy a new bike which he uses to return home. He averaged 10 kilometers faster on his bike than on foot. If his total trip took 1 hour and 20 minutes, what is his walking speed in kph? Use the formula.
Using the formula , we derive the formula for the time. The equation now becomes. Multiply both sides of the equation by the LCD and solve the resulting equation. E To solve rational inequalities: EP a Rewrite the inequality as a single rational expression on one side of the inequality symbol and 0 on the other side. Locate the x values for which the rational expression is zero or D undefined factoring the numerator and denominator is a useful strategy. Mark the numbers found in i on a number line.
Use a shaded circle to indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers partition the number line into intervals. Select a test point within the interior of each interval in ii. The sign of the rational expression at this test point is also the sign of the rational expression at each interior point in the aforementioned interval.
Summarize the intervals containing the solutions. Multiplying both sides of an inequality by a number requires that the sign positive or negative of the number is known. Since the sign of a variable is unknown, it is not valid to multiply both sides of an inequality by a variable.
Solve the inequality. Mark these on the number line. C D c Choose convenient test points in the intervals determined by —1 and 1 to determine the sign of in these intervals. Construct a table of signs as E shown below. Plot this set on the number line. Plot these points on a number line.
Use hollow circles since these values are not part of the solution. O c Construct a table of signs to determine the sign of the function in each interval determined by —1, 0, and 2. The solution set of the inequality is the set. A box with a square base is to have a volume of 8 cubic meters.
Let x be the length of the side of the square base and h be the height of the box. What are the possible measurements of a side of the square base if the height should be longer than a side of the square base?
The volume of a rectangular box is the product of its width, length, and height. Since the base of the box is square, its width and length are equal. The variable x is the length of a side of the box, while h is its height. The equation relating h and x is. Expressing h in terms of x, we obtain Since the height is greater than the width, and our inequality is. Plot C on a number line and use hollow circles since these values are not part of the solution.
D c Construct a table of signs to determine the sign of the function in each interval determined by 0 and 2. Note that is positive for any E real values of x. Therefore the height of the box should be less than 2 meters. A dressmaker ordered several meters of red cloth from a vendor, but the vendor only had 4 meters of red cloth in stock.
The vendor bought the remaining lengths of red cloth from a wholesaler for P1, He then sold those lengths of red cloth to the dressmaker along with the original 4 meters of 30 All rights reserved.
If the vendor's price per meter is at least P Let the variable x be the length of the additional cloth purchased by the vendor from the wholesaler. The wholesaler's price of red cloth per meter can be expressed as. If the vendor sold his cloth to the dressmaker at a price that is at least P Plot on a number line and use hollow circles since these values are not part of the solution set. The figure below is not drawn to scale. However, since we are dealing with lengths of cloth, we discard the interval where the length is negative.
Therefore the vendor bought PY and sold an additional length of red cloth from 16 — 28 meters to the dressmaker. Solved Examples O 1. Solve for x:. The LCD is C. Plot the points on a number line and use hollow circles since these values are not part of the solution set.
The solution set is given by. The graph is shown below. Solve for x: D Solution. Mark these on the number line where is included while the others are not. Solve for x: D 2. Solve for x: 3. Solve for x: E 4. Solve for x: EP 5. Solve for x: 6. If a and b are real numbers such that , find the solution set of. Two ships traveling from Dumaguete to Cagayan de Oro differ in average speed by 10 kph.
The slower ship takes 3 hours longer to travel a kilometer route than for the faster ship to travel a kilometer route.
Find the speed of the slower ship. Lesson 7: Representations of Rational Functions Learning Outcome s : At the end of the lesson, the learner is able to represent a rational function through its table of values, graphs and equation, and solve problems involving rational functions.
Table of values, graphs and equations as representations of a rational function. Rational functions as representations of real-life situations PY Definition: A rational function is a function of the form where and are polynomial functions and is not the zero polynomial i. The domain of is all values of where. Average speed or velocity can be computed by the formula. Consider a O meter track used for foot races. The speed of a runner can be computed by taking the time for him to run the track and applying it to the formula C , since the distance is fixed at meters.
Represent the speed of a runner as a function of the time it takes to run meters in the track. Since the speed of a runner depends on the time it takes to run meters, we can represent speed as a function of time. E Let x represent the time it takes to run meters. Then the speed can be represented as a function as follows: EP Observe that it is similar to the structure to the formula relating speed, distance, and time.
Continuing the scenario above, construct a table of values for the speed of a runner against different run times. A table of values can help us determine the behavior of a function as the variable changes. The current world record as of October for the meter dash is 9. We start our table of values at 10 seconds. Let x be the runtime and be the speed of the runner in meters per second, where. The table of values for run times from 10 seconds to 20 seconds is as follows: 10 12 14 16 18 20 10 8.
We can use a graph to determine if the points on the function follow a smooth curve or a straight line. Plot the points on the table of values on a Cartesian plane. PY Determine if the points on the function follow a smooth curve or a straight line. Assign points on the Cartesian plane for each entry on the table of values above: O A 10,10 B 12,8. PY For the meter dash scenario, we have constructed a function of speed against time, and represented our function with a table of values and a graph.
O The previous example is based on a real world scenario and has limitations on the values of the x-variable. For example, a runner cannot have negative time which would mean he is running backwards in time! However, we can apply the skills of constructing tables of values and plotting graphs to observe the behavior of rational functions. D Example 4. Represent the rational function given by using a table of values and plot a graph of the function by connecting points.
Since we are now considering functions in general, we can find function values across more values of x. Let us construct a table of values for EP some x-values from to 0 2 4 6 8 10 1. Connecting the points on this graph, we get: Why would the graph unexpectedly break the smooth curve and jump from point E to point F?
Let us take a look at the function PY again: Observe that the function will be undefined at. This means that there O cannot be a line connecting point E and point F as this implies that there is a point in the graph of the function where.
We will cover this aspect of graphs of rational functions in a future lesson, so for now we just present a partial graph for the function above as follows: C E D EP D Example 5. Represent the rational function using a table of values. Plot the points given in the table of values and sketch a graph by connecting the points.
As we have seen in the previous example, we will need to take a look at the x-values which will make the denominator zero. In this function, will make the denominator zero. Taking function values for integers in we get the following table of values: 2 3 4 5 6 7 8 9 10 0 6 0 1. Plotting the values above as points in the Cartesian plane: PY O We connect the dots to sketch the graph, but we keep in mind that is not C part of the domain.
For now we only connect those with values and those with values E D EP D Note that and are zeroes of the rational function, which means that the function value at these values is zero. These x-values give the x-intercepts of the graph. The behavior of the function near those values which make the function undefined will be studied in the next few lessons.
What will be their winning percentage if they win: a 10 games in a row? Let be the number of wins the Barangay Culiat needs to win in a row.
PY Then the percentage is a function of the number of wins that the team needs to win. The function can be written as: Construct a table of values for : O 10 15 20 30 50 0. Ten goats were set loose in an island and their population growth can be approximated by the function where P represents the goat population in year t since they were set loose. Recall that the symbol denotes the greatest integer function. The model suggests that the island can only support up to 59 goats. Note that since the model is just an approximation, there may be errors and the number 59 may not be exact.
O Solved Examples 1. Given , C a Construct a table of values using the numbers from to. Connecting the points, we get the following graph which forms two different smooth curves. Using integer values from C to 5, find the interval where the smooth curve of the following functions will disconnect: a D b E Solution.
So we can say that it disconnects at the interval b 0 1 2 3 4 5 und 0 0. It disconnects at the interval. It shoots up at but it starts decreasing after that. EP Lesson 7 Supplementary Exercises 1. Construct a table of values for the following functions using the integers from to. Using the table of values you got from the previous question, plot and connect the points of c e 3. A certain invasive species of fish was introduced in a small lake and their population growth can be modeled with time by the function a Construct a table of values b Is their population approaching a specific value?
Lesson 8: Graphing Rational Functions Learning Outcome s : At the end of the lesson, the learner is able to find the domain and range, intercepts, zeroes, asymptotes of rational functions, graph rational functions, and solve problems involving rational functions.
Domain and range of rational functions. Intercepts and zeroes of rational functions. Vertical and horizontal asymptotes of rational functions. Graphs of rational functions PY Recall: a The domain of a function is the set of all values that the variable x can take.
O c The zeroes of a function are the values of x which make the function zero. The real numbered zeroes are also x-intercepts of the graph of the function. Consider the function C a Find its domain, b intercepts, c sketch its graph and d determine its range. In addition, other values of x will make the function undefined. EP b The x-intercept of f x is 2 and its y-intercept is —1. Recall that the x-intercepts of a rational function are the values of x that will make the function zero.
A rational function will be zero if its numerator is zero. Therefore the zeroes of a rational function are the zeroes of its numerator. Since it is a real zero, it is also an x-intercept. The y-intercept of a function is equal to f 0. In this case,. Recall that in the previous lesson, we simply skipped connecting the points at integer values. Let us see what happens when x takes on values that brings the denominator closer to zero. Let us look at the values of x close to —2 on its left side i.
Table of values for x approaching —2— —3 —2. PY Notation. We call this line a vertical asymptote, formally defined as follows: Definition. We will also look how the function behaves as x increases or decreases without bound. PY iii. Table of values for f x as 5 10 1, 10, As 0. Table of values for f x as —5 2. Observe that as x increases or decreases without bound, f x gets closer and closer to 1.
We call this line a horizontal asymptote, formally defined as follows: 46 All rights reserved. A rational function may or may not cross its horizontal asymptote. PY Construct a table of signs to determine the sign of the function on the intervals determined by the zeroes and the intercepts.
D Plot the zeroes, y-intercept, and the asymptotes. Draw a short segment across 2,0 E to indicate that the function transitions from negative to positive at this point. EP We also know that f x increases without bound as and f x decreases without bound as as. Sketch some arrows near the asymptote to indicate this information. D Figure 2. Trace the arrowheads along with the intercepts using smooth curves. Do not cross the vertical asymptote. PY Figure 2. Therefore only the value 1 is not included in the range of f x.
The range of f x is. Find the horizontal asymptote of. We have seen from the previous example that the horizontal asymptotes can be determined by looking at the behavior of rational functions when x is very large i.
However, at extreme values of x, the value of a polynomial can be approximated using the value of the leading term. A good approximation is the value of , which is 4,, Similarly, for extreme values of , the value of can be approximated by.
Thus, for extreme values of , then can be approximated by , and therefore approaches 4 for extreme values of. PY Example 3. Following the idea from the previous example, the value of can O be approximated by for extreme values of.
Thus, the horizontal asymptote is C. Again, based on the idea from the previous example, the value of E can be approximated by for extreme values of. EP If we substitute extreme values of in , we obtain values very close to 0. Show that can be approximated by. If we substitute extreme values of in , we obtain extreme values as well. Thus, if takes on extreme values, then y also takes on extreme values and does not approach a particular finite number.
The function has no horizontal asymptote. We summarize the results from the previous examples as follows: Finding the Horizontal Asymptotes of a Rational Function Let be the degree of the numerator and be the degree of the denominator.
O x-intercept Find the values of x where the numerator will be zero. Find the values of a where the denominator is zero. Sketch the graph of. Find its domain and range. The numerator and denominator of f x can be factored as follows: 50 All rights reserved. Draw sections of the graph near the asymptotes based on the transition indicated on the table of signs. PY O Figure 2. Complete the sketch by connecting the arrowheads, making sure that the sketch C passes through the y-intercept as well.
The sketch should follow the horizontal asymptote as the x-values goes to the extreme left and right of the Cartesian plane. O The domain of the function is all values of x not including those where the function is undefined.
Therefore the domain of f x is C From the graph of the function, we observe that the function increases and decreases without bound. The graph also crosses the horizontal asymptote. Therefore the range of the function is the set of all real numbers. D Solved Examples E 1.
Let a Find its domain, b intercepts, c asymptotes. Next, d sketch its graph and e determine its range. The degree of the numerator is equal to the D degree of the denominator.
Find its domain, b intercepts, c asymptotes. Next, d D sketch its graph. EP a The domain of f x is. The degree of the numerator is less than the degree of the denominator. D d The table of signs is shown below. Calculus is needed to determine the range of this function. Past records from a factory suggest that new employees can assemble N t components per day after t days of being on the job, where. Sketch the graph of N.
Identify the horizontal asymptote of N, and discuss its meaning in practical terms. D a The domain of N t , as stated in the problem, is. Negative values of t are not allowed because t refers to a number of days.
E c There is no vertical asymptote in the stated domain. The degree of the numerator and denominator are equal. EP d The table of signs is shown below. Lesson 8 Supplementary Exercises 1. Find all asymptotes of. Explain why the function is not asymptotic to the line. Sketch the graph of this function. Sketch the graph of and give its domain, intercepts, asymptotes, and range.
Sketch the graph of and give its domain, intercepts, and asymptotes. Sketch the graph of c. Identify the horizontal asymptote of c, and discuss its meaning in practical terms. A challenging riddle. E EP D 57 All rights reserved. Lessons 1- 8 Topic Test 1 1. True or False [6] a A function is a set of ordered pairs such that no two ordered pairs have the same -value but different -values b The leading coefficient of is 3. Given , what is? Given and , find: [15] a b O c 5.
Is the solution set of in set builder notation C? Identify the zeroes of the function. For what values will the function be undefined? Identify the asymptotes of the graph below. Lessons 1 — 8 Topic Test 2 1. A part-time job gives you an hourly wage of P If you work for more than 40 hours per week, you get an overtime pay that is 1. Write a piecewise function that gives your weekly pay in terms of the number of hours you worked that week.
Given the piecewise function , evaluate the function at the following values of x: [5] PY a b c d O 3. Let , , and , find. Solve for x: C [10] D 5. Give the solution set of in set builder notation. Find the asymptotes of. Lesson 9: One-to-One functions Learning Outcome s : At the end of the lesson, the learner is able to represent real- life situations using one-to-one functions.
One-to-one functions 2. Examples of real-life situations represented by one-to-one functions. Horizontal line test. That is, the same -value is never paired with two different -values. In Examples , determine whether the given relation is a function.
If it is a function, determine whether it is one-to-one or not. O Example 1. Thus, the C relation is a function. Further, two different members cannot be assigned the same SSS number.
Thus, the function is one-to-one. The relation pairing a real number to its square. Each real number has a unique perfect square. Thus, the relation is a function. However, two different real numbers such as 2 and —2 may have the same E square.
Thus, the function is not one-to-one. The relation pairing an airport to its airport code EP Airport codes are three letter codes used to uniquely identify airports around the world and prominently displayed on checked-in bags to denote the destination of these bags.
Since each airport has a unique airport code, then the relation is a function. Also, since no two airports share the same airport code, then the function is one-to-one. The relation pairing a person to his or her citizenship. The relation is not a function because a person can have dual citizenship i.
Example 5. The relation pairing a distance d in kilometers traveled along a given jeepney route to the jeepney fare for traveling that distance. The relation is a function since each distance traveled along a given jeepney route has an official fare. In fact, as shown in Lesson 1, the jeepney fare may be represented by a piecewise function, as shown below: PY Note that is the floor or greatest integer function applied to.
However, the function is not one-to-one because different distances e. That is, because , then F is not one-to-one. O A simple way to determine if a given graph is that of a one-to-one function is by using the Horizontal Line Test.
C Horizontal Line Test. A function is one-to-one if each horizontal line does not intersect the graph at more than one point. A graph showing the plot of fails the horizontal line test because some D lines intersect the graph at more than one point. The Vertical and Horizontal Line Tests. All functions satisfy the vertical line test. E All one-to-one functions satisfy both the vertical and horizontal line tests.
Solved Examples EP 1. Which of the following are one-to-one functions? Only b is a one-to-one function. Books can have multiple authors that wrote the book. Which of the following relations is a one-to-one function? Both a and c are one-to-one functions. B is a function however it is not one-to-one since it has y-values that are paired up with two different x-values.
Lesson 9 Supplementary Exercises 1. Consider each uppercase letter in the English alphabet as a graph. Is there any of these letters that will pass both the vertical and horizontal line tests? The length of a rectangle, , is four more than its width. Let be the function PY mapping the length of the rectangle to its area.
Is the function one-to-one? Lesson Inverse of One-to-One Functions Learning Outcome s : At the end of the lesson, the learner is able to determine the O inverses of one-to-one functions. Lesson Outline: C 1. Inverse of a one-to-one function. Finding the inverse of a one-to-one function. Property of inverse functions D The importance of one-to-one functions is due to the fact that these are the only E functions that have an inverse, as defined below.
Definition: Let f be a one-to-one function with domain A and range B. Then the EP inverse of f, denoted by f—1, is a function with domain B and range A defined by if and only if for any y in B.
A function has an inverse if and only if it is one-to-one. If a function f is not one- to-one, properly defining an inverse function f—1 will be problematic.
If f —1 exists, then f—1 5 has to be both 1 and 3, and this prevents f—1 from being a valid function. This is the reason why the inverse is only defined for one-to-one functions. Find the inverse of Solution. The equation of the function is Interchange the x and y variables: 62 All rights reserved. Solve for y in terms of x: Therefore the inverse of is. O C For the second and third properties above, it can be imagined that evaluating a function and its inverse in succession is like reversing the effect of the function.
For example, the inverse of a function that multiplies 3 to a number and adds 1 is a function that subtracts 1 and then divides the result by 3. Find the inverse of. The equation of the function is. Interchange the x and y variables:. EP Solve for y in terms of x: D The inverse of is. Find the inverse of the rational function. The equation of the function is: Interchange the x and y variables : 63 All rights reserved. Solve for y in terms of x: Place all terms with y on one side and those without y on the other side.
PY Therefore the inverse of is. O Example 4. Find the inverse of , if it exists. The students should recognize that this is a quadratic function with a C graph in the shape of a parabola that opens upwards. It is not a one-to-one function as it fails the horizontal line test.
Optional We can still apply the procedure for finding the inverse of a one-to-one D function to see what happens when it is applied to a function that is not one-to-one. The equation of the function is: E Interchange the x and y variables: Solve for y in terms of x EP D Complete the square The equation does not represent a function because there are some x-values that correspond to two different y-values e. Therefore the function has no inverse function. This function fails the horizontal line test and therefore has no inverse.
Alternate Solution. We can also show that f—1 does not exist by showing that f is not one-to-one. Since the x-values 1 and —1 are paired to the same y-value, then f is not one-to-one and it cannot have an inverse. Therefore f x has no EP inverse function. To convert from degrees Fahrenheit to Kelvin, the function is , where t is the temperature in Fahrenheit Kelvin is the SI unit of D temperature.
Find the inverse function converting the temperature in Kelvin to degrees Fahrenheit. The equation of the function is: , Since k and t refer to the temperatures in Kelvin and Fahrenheit respectively, we do not interchange the variables.
Solve for t in terms of k: Therefore the inverse function is where k is the PY temperature in Kelvin. C E D Therefore,. D Therefore,. Lesson 10 Supplementary Exercises 1. Which among the following functions have an inverse? Find if. Lesson Graphs of Inverse Functions PY Learning Outcome s : At the end of the lesson, the learner is able to represent an inverse function through its table of values and graph, find the domain and range of an inverse function, graph inverse functions, solve problems involving inverse functions.
O Lesson Outline: C 1. Domain and range of a one-to-one function and its inverse D Graphing Inverse Functions First we need to ascertain that the given graph corresponds to a one-to-one function E by applying the horizontal line test. If it passes the test, the corresponding function is one-to-one.
EP Given the graph of a one-to-one function, the graph of its inverse can be obtained by reflecting the graph about the line. Graph if the graph of restricted in the domain is given below. What is the range of the function? What is D the domain and range of its inverse? Take the reflection of the restricted graph of across the line.
PY The range of the original function can be determined by the inspection of the graph. The range is. Verify using techniques in an earlier lesson that the inverse function is given by O.
Is this true for all one- to-one functions and their inverses? Find the inverse of using its given graph. D 68 All rights reserved. Applying the horizontal line test, we verify that the function is one-to-one. Since the graph of is symmetric with respect to the line indicated by a dashed line , its reflection across the line is itself.
Therefore the inverse of is itself or. PY Verify that using the techniques used in the previous lesson. O Example 3. Find the inverse of using the given graph. Applying the horizontal line test, we confirm that the function is one-to-one. Reflect the graph of across the line to get the plot of the inverse function.
D The result of the reflection of the graph of is the graph of Therefore,. D b Find the equation of its asymptotes. EP a From our lessons on rational functions, we get the following results: Domain of Range of D b Using techniques from the lesson on rational functions, the equations of the asymptotes are Vertical asymptote: Horizontal asymptote: 70 All rights reserved.
In fact, the asymptotes could also be obtained by reflecting the D original asymptotes about the line. Vertical asymptote: Horizontal asymptote: E d The domain and range of the functions and its inverse are as follows: EP Domain Range We can make the observation that the domain of the inverse is the range of the D original function and the range of the inverse is the domain of the original function.
In the examples above, what will happen if we plot the inverse functions of the inverse functions? If we plot the inverse of a function, we reflect the original function about the line. If we plot the inverse of the inverse, we just reflect the graph back about the line and end up with the original function. This result implies that the original function is the inverse of its inverse, or.
Solving problems involving inverse functions We can apply the concepts of inverse functions in solving word problems involving reversible processes. You asked a friend to think of a nonnegative number, add two to the number, square the number, multiply the result by 3 and divide the result by 2. If the result is 54, what is the original number? Construct an inverse function that will provide the original number if the result is given.
We first construct the function that will compute the final number based on PY the original number. Following the instructions, we come up with this function: O The graph is shown below, on the left. This is not a one-to-one function because the graph does not satisfy the horizontal line test. However, the instruction indicated that the original number must be nonnegative. The domain of the function must thus be restricted to C , and its graph is shown on the right, below.
Interchange the x and y variables: 72 All rights reserved. Solve for y in terms of x: Since we do not need to consider PY Finally we evaluate the inverse function at to determine the original number: O The original number is 4. Engineers have determined that the maximum force in tons that a C particular bridge can carry is related to the distance in meters between it supports by the following function: D How far should the supports be if the bridge is to support 6.
Construct an inverse function to determine the result. EP To lessen confusion in this case, let us not interchange and as they denote specific values. Solve instead for in terms of : D The inverse function is. The supports should be placed at most 6. If is restricted on the domain , what is the domain of its inverse? The domain of the inverse of is just the range of.
Therefore the domain of is 2. Given the graph of below, sketch the graph of its inverse. So we get: EP D 74 All rights reserved. Using algebraic methods, construct the inverse of. Is the function you get the same as the sketch of the inverse in the previous number? To get , we first interchange x and y in. So we get. We then isolate y So we get. However, the graph of that will result in a parabola PY opening downwards while the sketch we have in number 2 was just half that parabola.
This occurs because the function must be one-to-one to have an inverse. Lesson 11 Supplementary Exercises O 1. Find the domain and range of the inverse of with domain restriction. Give the vertical and horizontal asymptotes of. Give the vertical and horizontal asymptotes of its inverse. At what point will the graph of and its inverse intersect? The formula for converting Celsius to Fahrenheit is given as where C is the temperature in Celsius and F is the temperature in Fahrenheit.
Find the formula for converting Fahrenheit to Celsius. If the temperature in a EP thermometer reads Explain why the function is one-to-one, even if it is a quadratic function. Find the inverse of this function D and approximate the length of a single fish if its weight is grams. Lessons 9 — 11 Topic Test 1 1. True or False [6] a A linear function is a one-to-one function. Identify if the following are one-to-one functions or not. Which of the following functions have an inverse function?
If so, find its inverse. Sketch the graph of the inverse of the function. Find the inverse of the following functions: [15] a E b 2. Find the domain and range of the inverse of [10] EP 3. Find the asymptotes of the inverse of [10] 4. Lesson Representing Real-Life Situations Using Exponential Functions Learning Outcome s : At the end of the lesson, the learner is able to represent real- life situations using exponential functions. Exponential functions 2.
Population, half-life, compound interest 3. A better approximation is Let b be a positive number not equal to 1. Some of the most common applications in real-life of exponential functions and their transformations are population growth, exponential decay, and compound interest. Suppose that the bacteria doubles every hours. Give an exponential model for the bacteria as a function of t.
C The half-life of a radioactive substance is the time it takes for half of the substance to decay. Suppose that the half-life of a certain radioactive substance is 10 days D and there are 10g initially, determine the amount of substance remaining after 30 days, and give an exponential model for the amount of remaining substance.
We use the fact that the mass is halved every 10 days from definition of half-life. A starting amount of money called the principal can be invested at a certain interest rate that is earned at the end of a given period of time such as one year. If the interest rate is compounded, the interest earned at the end of the period is 78 All rights reserved. The same process is repeated for each succeeding period: interest previously earned will also earn interest in the next period.
De la Cruz invested P, Define an exponential model for this situation. How much will this investment be worth at the end of each year for the next five years? The investment is worth O P, Compound Interest. Referring to Example 5, is it possible for Mrs. De la Cruz to double her money in 8 years? The Natural Exponential Function D While an exponential function may have various bases, a frequently used based is the irrational number e, whose value is approximately 2.
The enrichment in Lesson 27 will show how the number e arises from the concept of compound interest. Because e is a commonly used based, the natural exponential function is defined having e as the base.
A large slab of meat is taken from the refrigerator and placed in a pre- heated oven. Construct a table of values for the following values of t: 0, 10, 20, 30, 40, 50, 60, and interpret your results. Round off values to the nearest integer. Robert invested P30, after graduation.
If the average interest rate is 5. The money has more than doubled in 15 years. There will be bacteria after 40 days. The half-life of a substance is years. Use this model to approximate the Philippine population during the years , , , and Round of answers to the nearest thousand. A barangay has 1, individuals and its population doubles every 60 years. Give an exponential model for the barangay. E Give an exponential model for a sum of P10, invested under this scheme.
How much money will there be in the account after 20 years? The half-life of a radioactive substance is years. If the initial amount of the substance is grams, give an exponential model for the amount remaining after t years.
What amount of substance remains after years? D 81 All rights reserved. Lesson Exponential Functions, Equations, and Inequalities Learning Outcome s : At the end of the lesson, the learner is able to distinguish among exponential functions, exponential equations and exponential inequality.
PY The definitions of exponential equations, inequalities and functions are shown below. An exponential function expresses a relationship between two variables such as x and y , and can be represented by a table of values or a graph Lessons 14 and E Solved Examples EP Determine whether the given is an exponential function, an exponential equation, an exponential inequality, or none of these. Lesson Solving Exponential Equations and Inequalities Learning Outcome s : At the end of the lesson, the learner is able to solve exponential equations and inequalities, and solve problems involving exponential equations and inequalities Lesson Outline: 1.
Solve exponential equations 2. Write both sides with 4 as the base. Write both sides with 2 as the base. Both and 25 can be written using 5 as the base.
Solve the equation. Both 9 and 3 can be written using 3 as the base. E EP Example 4. The half-life of Zn is 2. How much time has passed? Using exponential models in Lesson 12, we can determine that after t EP minutes, the amount of Zn in the substance is. We solve the equation. Solved Examples Solve for x in the following equations or inequalities.
How much time will have elapsed when only 15 grams remain? The amount of substance after t hours. Lesson 14 Supplementary Exercises In Exercises , solve for x.
How much time is needed for a sample of Pd to lose Pd has a half-life of 3. A researcher is investigating a specimen of bacteria. She finds that the original bacteria grew to 2,, in 60 hours. How fast does the bacteria a double? Lesson Graphing Exponential Functions Learning Outcome s : At the end of the lesson, the learner is able to represent an exponential function through its a table of values, b graph, and c equation, find the domain and range of an exponential function, determine the intercepts, zeroes, and asymptotes of an exponential function, and graph exponential functions Lesson Outline: 1.
Domain, range, intercepts, zeroes, and asymptotes. In the following examples, the graph is obtained by first plotting a few points. Results PY will be generalized later on. O Step 1: Construct a table of values of ordered pairs for the given function.
As x decreases without bound, the function approaches 0, i. As x increases without bound, the function approaches 0, i. The domain is the set of all real numbers. The range is the set of all positive real numbers. It is a one-to-one function. It satisfies the Horizontal Line Test. The y-intercept is 1. Phone: Toll free: III Mathematical symbols. At that the close of the novel a listing of applicable formulae contained inside the text is included for convenience of reference.
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