1

Common Core Math II Unit 3B

Inverse Functions

Logarithms

2

Inverse of a Function In science, you may have converted from degrees Fahrenheit to degrees Celsius. The standard formula to convert from Fahrenheit to Celsius is: The function f(C) is the process for converting degrees Celsius to degrees Fahrenheit.

a) Write the function f (C)

b) What is the value of f (25)? What does it represent?

c) Solve the equation for f(C) = 112 and describe the meaning of your answer.

d) What happens if you want to input degrees Fahrenheit and output degrees Celsius? Reverse the process of the formula by solving for C and name this function g(F).

e) What is g (f (x))?

325

9)( CCfF

where C is a number of degrees Celsius and f(C) is number of degrees Fahrenheit.

3

The word “inverse” is used in several different ways in mathematics. For example, we say that -6 is the additive inverse of 6 because -6 + 6 = 0. Essentially adding the inverse of -6 has the effect of undoing the addition of 7. You can think of this property as a way of retrieving the original number. Similarly, we say that (7) is the multiplicative inverse of (1/7) because (7/1)(1/7) = 1. Multiplying by 7 and then by (1/7) has the effect of undoing the multiplication by 7 and retrieving the original number. Look at the situation below: Kathy and Kevin are sharing their graphs for the same set of data. Both students insist that they are correct, but yet their graphs are different. They have checked and re-checked their data and graphs. Can you explain what might have happened? Has this ever happened to you? Kathy’s Graph Kevin’s Graph

You are correct if you thought the independent and dependent variables were just reversed. In mathematics, when the independent and dependent variables are reversed, we have what is called the inverse of the function. If you look at Kathy and Kevin’s graph again, you might notice that the ordered pairs have been switched. For example in Kathy’s graph the ordered pairs of (0,1) and (2,4) are (1,0) and (4,2) in Kevin’s graph. We talked about inverse operations above and how they help to undo the operation. The inverse of a function helps to get back to an original value of x. Do all functions have inverses? Sometimes this is best explained by looking at a mapping of functions, as shown in the illustration below.

Does f(x) = x have an inverse? In the mapping, does an input of -3 always give an output of 3? Does an output of 3 always come from an input of -3? How does the diagram show reasons for your answer?

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According to the mapping on the left, the output of 3 has two possible inputs (3 and -3). The function f(x) = x does not have an inverse since several input values do not have their own “private” output value. In order to have an inverse we should be able to simply switch the x and y values in the ordered pairs, as Kathy and Kevin did with their graphs. Does g(x) = x + 3 have an inverse? How does the diagram show reasons for your answer? Let’s see, we have the ordered pairs (1,3), (2,4), and (3,6). The inverse of those ordered pairs are (3,1), (4,2), and (6,3). All you have to do is just reverse the direction of the arrow, so g(x) does have an inverse. How can you use a mapping diagram like those shown above to decide whether a function does or does not have an inverse function? The key to deciding whether a function does or does not have an inverse is whether each x is mapped to its own “private” range element. If a function has an inverse, there will never be two range elements. A mathematical function f sets up a correspondence between two sets so that each element of the domain D is assigned exactly one image in the range R. If another function g makes assignments in the opposite direction so that when f(x) = y then g(y) = x, we say that g is the inverse of f. The inverse relationship between two functions is usually indicated with the notation g = f-1 or g(x) = f-1(x). The notation f-1 is read “f inverse of x.” In this context, the exponent “-1” does not mean reciprocal “one over f(x).” Inverses are often useful in solving problems, but there are many functions that do have inverses. For functions with numeric domains and ranges, it is usually helpful to describe the rule of assignment with an algebraic expression. It is also useful to find such rules for inverse function. As we work through the next investigation, we will try to answer the following questions: Which types of functions have inverses? How can rules for inverse functions be found?

Investigation: Inverses Consider the following functions: 1) f(x) = 6 + 3x|

2) f(x) =

Part 1: Graphs of Inverse Functions For each of the functions above, follow steps 1 – 4.

1) Make a table of 5 values and graph the function (X1, Y1). 2) Complete X2 & Y2 by switching the X1 and Y1 values and graph the inverse on the same coordinate plane. 3) What do you notice about the two graphs? 4) What line are the inverses reflected over?

X1 Y1 X2 Y2

X1 Y1 X2 Y2

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Part 2: Equations of Inverse Functions We saw in part 1 of the investigation that functions 1 and 2 are inverse functions. We also know that we can find inverses of tables by switching the x and y values in a table. So the question we want to explore now is how to find the equation of an inverse function. For each of the functions above, follow steps 5 – 6.

5) Take the function and switch the x and y values. 6) Then solve for y.

The equation for the function one’s inverse should be function 2 and the inverse for function 2 should be the equation for function 1. This process will work for any function which has an inverse. So, let’s try some different types in the problems below. NOW TRY Graph y = x2 + 3 and find the inverse by interchanging the x and y values of several ordered pairs. Is the inverse a function? Check by graphing both y = x2 + 3 and the inverse on the graph on the right. The graph of y = x2 + 3 is a parabola that opens upward with vertex (0, 3). The reflection of the parabola in the line y = x is the graph of the inverse. The inverse is a parabola with a vertex at (3, 0) that opens to the right. Since this “sideways” parabola does not pass the vertical line test, this inverse is not a function. However, if you consider half of the parabola, then this function has an inverse. What?? Let’s look at a simpler function y = x2. If you only consider the positive values of x in the original function, the graph is half of a parabola. When you reflect the “half” over the y = x line, it will pass the vertical line test as shown on the right. Algebraically, let’s switch the x and y values and solve for y. Function: y = x2, where x ≥ 0 Switch the x and y values: x = y2

Solve for y: y = As shown, the both graphs pass the vertical line tests and are inverse functions of each other. The above example means that if you restrict the domains of some functions, then you will be able to find inverse functions of them.

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NOW TRY Find the inverses of the functions below. Graph the function and it’s inverse on graph paper.

1) f(x) = x3 2) y = -3x + 4

3) f(x) = 4) y = 2x

©8 b2B0Z162E 9KeuWtUa2 7Sqozfst6wlaWrveH ELQLsC0.x p UANlGlB brxigghhdtysT qr3eTsmefrzvWeEdj.6 O oMraDdGeH jwxiNtPhp OIFnSf6iwnMiKtKeG RAFlcgTeZbEr0aS 22W.d Worksheet by Kuta Software LLC

Kuta Software - Infinite Algebra 2 Name___________________________________

Period____Date________________Function Inverses

State if the given functions are inverses.

1)

g(

x) =

4 -

3

2

x

f (

x) =

1

2

x +

3

2

2)

g(

n) =

-12 - 2

n

3

f (

n) =

-5 + 6

n

5

3)

f (

n) =

-16 +

n

4

g(

n) =

4

n + 16

4)

f (

x) =

-4

7

x -

16

7

g(

x) =

3

2

x -

3

2

5)

f (

n) =

-

(

n + 1)3

g(

n) =

3 +

n3

6)

f (

n) =

2

(

n - 2)3

g(

n) =

4 + 3

4

n

2

7)

f (

x) =

4

-

x - 2 + 2

h(

x) =

-1

x + 3

8)

g(

x) =

-2

x - 1

f (

x) =

-2

x + 1

Find the inverse of each function.

9)

h(

x) =

3

x - 310)

g(

x) =

1

x - 2

11)

h(

x) =

2

x3 + 3 12)

g(

x) =

-4

x + 1

-1-

7

©8 b2B0Z162E 9KeuWtUa2 7Sqozfst6wlaWrveH ELQLsC0.x p UANlGlB brxigghhdtysT qr3eTsmefrzvWeEdj.6 O oMraDdGeH jwxiNtPhp OIFnSf6iwnMiKtKeG RAFlcgTeZbEr0aS 22W.d Worksheet by Kuta Software LLC

Kuta Software - Infinite Algebra 2 Name___________________________________

Period____Date________________Function Inverses

State if the given functions are inverses.

1)

g(

x) =

4 -

3

2

x

f (

x) =

1

2

x +

3

2

2)

g(

n) =

-12 - 2

n

3

f (

n) =

-5 + 6

n

5

3)

f (

n) =

-16 +

n

4

g(

n) =

4

n + 16

4)

f (

x) =

-4

7

x -

16

7

g(

x) =

3

2

x -

3

2

5)

f (

n) =

-

(

n + 1)3

g(

n) =

3 +

n3

6)

f (

n) =

2

(

n - 2)3

g(

n) =

4 + 3

4

n

2

7)

f (

x) =

4

-

x - 2 + 2

h(

x) =

-1

x + 3

8)

g(

x) =

-2

x - 1

f (

x) =

-2

x + 1

Find the inverse of each function.

9)

h(

x) =

3

x - 310)

g(

x) =

1

x - 2

11)

h(

x) =

2

x3 + 3 12)

g(

x) =

-4

x + 1

-1-

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13)

g(

x) =

7

x + 18

2

14)

f (

x) =

x + 3

15)

f (

x) =

-

x + 3 16)

f (

x) = 4

x

Find the inverse of each function. Then graph the function and its inverse.

17)

f (

x) =

-1 -

1

5

x

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

18)

g(

x) =

1

x - 1

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

19)

f (

x) =

-2

x3 + 1

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

20)

g(

x) =

-

x - 5

3

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

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13)

g(

x) =

7

x + 18

2

14)

f (

x) =

x + 3

15)

f (

x) =

-

x + 3 16)

f (

x) = 4

x

Find the inverse of each function. Then graph the function and its inverse.

17)

f (

x) =

-1 -

1

5

x

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

18)

g(

x) =

1

x - 1

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

19)

f (

x) =

-2

x3 + 1

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

20)

g(

x) =

-

x - 5

3

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-2-

8

Independent Practice with Inverse Functions 1) The following graphs show pairs of functions that are inverses of each other and have been reflected

over the line y = x. For each pair of functions: Find two points (a, b) and (c, d) on one graph and show that the points (b, a) and (d, c) are on the

other graph. Explain why every point (x, y) on one graph maps to a point (y, x) on the other graph.

2) Which of these graphs represent functions that have inverses? Be prepared to justify each answer.

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3) Find the rules for the inverses of the following functions.

a. f(x) = 4x – 5

b. g(x) = 8x2 (domain x ≥ 0)

c. h(x) = 5/x (domain x ≠ 0)

d. k(x) = -5x + 7

4) For the following statements, write a rule and determine whether the rule has an inverse. a. If regular gasoline is selling for $3.95 per gallon, the price of any purchase p is a function of the

number of gallons of gasoline g in that purchase.

b. If a school assigns 20 students to each mathematics class, the number of mathematics classes M is

a function of the number of mathematics students s in that school.

c. The area of a square A is a function of the length of each side s.

10

Common Core Math II Name Date

Logarithmic Functions: Inverse of an Exponential Function

The Richter Scale The magnitude of an earthquake is a measure of the amount of energy released at its source. The Richter scale is an exponential measure of earthquake magnitude, as shown on the right. The magnitude increases per unit as the energy released increases by powers of ten. A simpler way to examine the Richter Scale is shown below. An earthquake of magnitude 5 releases about 30 times as much energy as an earthquake of magnitude 4. In 1995, an earthquake in Mexico registered 8.0 on the Richter scale. In 2001, an earthquake of magnitude 6.8 shook Washington state. Let’s compare the amounts of energy released in the two earthquakes.

For the earthquake in Mexico at 8.0 on the Richter Scale, the energy released is E 308 and for the earthquake

in Washington state, the energy released is E 306.8. A ratio of the two quakes and using the properties of exponents yields the following:

Mexico earthquake E 308 308

Washington earthquake E 306.8 306.8

308-6.8 = 301.2 59.2 What this means is that the earthquake in Mexico released about 59 times as much energy as the earthquake in Washington. The exponents used by the Richter scale shown in the above example are called logarithms or logs. A logarithm is defined as follows:

The logarithm base b of a positive number y is defined as follows: If y = bx, then logby = x.

Energy Released: X 30

0 1 2 3 6 5 4 9 8 7

Magnitude: +1

E E 301 E 302 E 303 E 307 E 306 E 305 E 304 E 308 E 309

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The exponent x in the exponential expression bx is the logarithm in the equation logby = x. The base b in bx is the same as the base b in the logarithm. In both cases, b ≠ 1 and b > 0. So what this means is that you use logarithms to undo exponential expressions or equations and you use exponents to undo logarithms, which means that the operations are inverses of each other. Thus, an exponential function is the inverse of a logarithmic function and vice versa.

Key Features of Logarithmic Graphs

A logarithmic function is the inverse of an exponential function. The graph show y = 10x and y = log x. Note that (0, 1) and (1,10) lie on the graph of y = 10x and that (1, 0) and (10, 1) lie on the graph of y = log x, which demonstrates the reflection over the line y = x. Since an exponential function y = bx has an asymptote at y = 0, the inverse function y = logbx has an asymptote at x = 0. The other key features of exponential and logarithmic functions are summarized in the box below.

Key Features of Exponential and Logarithmic Functions

Characteristic Exponential Function y = bx

Logarithmic Function y = logbx

Asymptote y = 0 x = 0

Domain All real numbers x> 0

Range y > 0 All real numbers

Intercept (0,1) (1,0)

Translations of logarithmic functions are very similar to those for other functions and are summarized in the table below.

Parent Function y = logbx

Shift up y = logbx + k

Shift down y = logbx - k

Shift left y = logb(x + h)

Shift right y = logb(x - h)

Combination Shift y = logb(x ± h) ± k

Reflect over the x-axis y = -logbx

Stretch vertically y = a logbx

Stretch horizontally y = logbax

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Let’s look at the following example. The graph on the right represents a transformation of the graph of f(x) = 3 log10 x + 1.

x = 3: Stretches the graph vertically. h = 0: There is no horizontal shift. k = 1: The graph is translated 1 unit up.

Now it’s your turn to find key features and translate logarithmic functions. TRY NOW Graph the following function on the graph at right. Describe each transformation, give the domain and range, and identify any asymptotes. y = -2log10(x + 2) – 4 Domain: Range: Asymptote: Description of transformations:

Guided Practice with Logarithmic Functions In 1812, an earthquake of magnitude 7.9 shook New Madrid, Missouri. Compare the amount of energy released by that earthquake to the amount of energy released by each earthquake below. 1) Magnitude 7.7 in San Francisco, California, in 1906.

2) Magnitude 3.2 in Charlottesville, Virginia, in 2001.

3) Magnitude 9.5 in Valdivia, Chile, in 1960.

4) Graph the exponential function and its inverse on graph at

right. Make a table if necessary.

Y = 2x Y = log2 x

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Graph the following transformations of the function y = log10 x on the coordinate planes. Determine the domain, range, and asymptotes of each transformation. Describe the transformations. 5) y = log10 x – 6 6) y = -log10 (x + 2) 7) y = log10 2x

Domain: Domain: Domain: Range: Range: Range: Asymptotes: Asymptotes: Asymptotes: Description: Description: Description:

Independent Practice with Logarithmic Functions

1. Describe in your own words what happens to the graph of f(x) = log2 x under the given transformations, then graph. a. f(x) = log2 (x – 2) b. f(x) = log2 (x) + 3 c. f(x) = log2 (x – 2) + 3

2. State the domain, range, intercepts and asymptotes of f(x) = log2 (x – 2) + 3. Domain: Range: Intercept(s): Asymptote(s):

3. Describe the transformations of y = 4 log2 (2x – 4) + 6 from the parent function y = log2 (x).

4. Describe the transformations of y = -3 log 10 (4x + 3) – 2 from the parent function y = log10 (x).

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Investigation 1: How Loud is too Loud?

Decibels and Sound Intensity

Use the table below to answer the Think About This Situation questions. Answer a)______________________________________________________________________ Answer b)______________________________________________________________________

Your analysis of the sound intensity data might have suggested several different algorithms for converting watts per cm2 into decibels. For example, if the intensity of a sound is 10x watts/cm2, its loudness in decibels is 10x + 120. The key to discovery of this conversion rule is the fact that all sound intensities were written as powers of 10. What would you have done if the sound intensity readings had been written as number like 3.45 watts/cm2 or 0.0023 watts/cm2?

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As you work on problems of this investigation, look for an answer to this question: How can any positive number be expressed as a power of 10?

1) Express each of the numbers below as accurately as possible as a power of 10. You can find exact values for some of the required exponents by thinking about the meanings of positive and negative exponents. Others might require some calculator exploration of ordered pairs that satisfy the exponential equation y = 10x. a. 100 b. 10,000 c. 1,000,000 d. 0.01 e. -0.001 f. 3.45 g. -34.5 h. 345 i. 0.0023

2) Suppose that the sound intensity of a screaming baby was measured as 9.5 watts/cm2. To calculate

the equivalent intensity in decibels, 9.5 must be written as 10x for some value of x. a. Between which two integers does it make sense to look for values of the required exponent? How

do you know?

b. Which of the two integer values in Part a is probably closer to the required power of 10?

c. Estimate the required exponent to the nearest hundredth. Then use your estimate to calculate a

decibel rating for the loudness of the baby’s scream.

d. Estimate the decibel rating for loudness of sound from a television set that registers intensity of 6.2 watts/cm2.

Common Logarithms As you discovered in your work on Problems 1 and 2, it is not easy to solve equations like 10x = 9.5 or 10x = 0.0023, even by estimation. To deal with this very important problem, mathematicians have developed procedures for finding exponents. We have already discussed logarithms in the previous lesson, but in this lesson we will solve exponent problems which focus on a base of 10. When we solve logarithmic problems in base 10, we call them common logarithms.

The definition of common logarithms is usually expressed as:

log10 a = b if and only if 10b = a

Log10 a is pronounced “log base 10 of a”. Because base 10 logarithms are so commonly used, log10 a is often written as log a. Most calculators have a built-in log function that automatically finds the required exponent value.

3) Use your calculator to find the following logarithms. Then compare the results with your work on

Problem 1. a. log 100 b. log 10,000 c. log 1,000,000 d. log 0.01 e. log (-0.001) f. log 3.45 g. log (-3.45) h. log 345 i. log 0.0023

16

4) What do your results from Problem 3 (especially Parts e and g) suggest about the kinds of numbers that have logarithms? See if you can explain your answer by using the connection between logarithms and the exponential function y = 10x.

5) Logarithms can be used to calculate the decibel rating of sounds, when the intensity is measured in watts/cm2. a. Use the logarithm feature of your calculator to rewrite 9.5 as a power of 10. That is, find x so that

9.5 = 10x.

b. Recall that if the intensity of a sound is 10x watts/cm2, then the expression 10x + 120 can be used to convert the sound’s intensity to decibels. Use your results from Part a to find the decibel rating of the crying baby in Problem 2.

6) Assume the intensity of a sound I = 10x watts/cm2. a. Explain why x = log I.

b. Rewrite the expression for converting sound intensity reading to decibel numbers using log I.

7) Use your conversion expression from Problem 6 to find the decibel rating of the television set in

Problem 2 Part d.

Why Do They Taste Different? You may recall from your study of science that the acidity of a substance is described by the pH rating – the lower the pH rating, the more acidic the substance is. The acidity depends upon the hydrogen ion concentration in the substance (in moles per liter). Some sample hydrogen ion concentrations are given below. Since those hydrogen ion concentrations are generally very small numbers, they are converted to the simpler pH scale for reporting. 8) Examine the table at right.

a. Describe how hydrogen ion concentrations [H+] are converted into pH readings.

b. Write an equation that makes use of logarithms expressing pH as a function of hydrogen ion concentration [H+].

9) Use the equation relating hydrogen ion concentration and pH reading to compare acidity of some

familiar liquids. a. Complete the table at right. Round results to the nearest tenth.

b. Explain how your results tell which is more acidic – lemonade,

apple juice, or milk.

17

A:___________________________________________________________________________ Bi.___________________________________________________________________________ Bii.__________________________________________________________________________ Biii.__________________________________________________________________________ Biv. __________________________________________________________________________

Use your understanding of the relationship between logarithms and exponents to help complete these tasks. a. Find these common (base 10) logarithms without using a calculator.

i. log 1,000 ii. log 0.001 iii. log 103.2

b. Use the function y = 10x, but not the logarithm key of your calculator to estimate each of these logarithms to the nearest tenth. Explain how you arrived at your answers. i. log 75 ii. log 750 iii. log 7.5

c. If the intensity of sound from a drag race car is 125 watts/cm2, what is the decibel rating of the loudness for that sound?

18

Independent Practice on Common Logarithms 1) Find the decibel ratings of these sounds.

a. A passing subway train with sound intensity reading of 10-0.5 watts/cm2.

b. An excited crowd at a basketball game with sound intensity reading of 101.25 5 watts/cm2.

2) Find these common logarithms without using a calculator and explain your reasoning.

a. Log 100,000 b. log 0.001 c. Log (104.74) d. log 1

3) Find the decibel rating of these sounds.

a. A door slamming with sound intensity 89 watts/cm2.

b. A radio playing with sound intensity 0.005 watts/cm2.

4) Pure water has a pH of 7. Liquids with pH less than 7 are called acidic; those with pH greater than 7

are called alkaline. Typical seawater has a pH about 8.5, soft drinks have pH about 3.1, and stomach gastric juices have pH about 1.7. a. Which of the three liquids are acidic and which are alkaline?

b. Find the concentration of hydrogen ions in seawater, soft drinks, and gastric juices.

c. Explain why it is correct to say that the concentration of hydrogen ions in gastric juices is about 25 times that of soft drinks.

d. If a new soft drink has a hydrogen ion concentration that is one-fifth that of typical soft drinks, what is its pH?

19

Investigation 2: Solving for Exponents Logarithms can be used to find exponents that solve equations like 10x = 9.5. For this reason, they are an invaluable tool in answering questions about exponential growth and decay. For example, the world population is currently about 6.2 billion and growing exponentially at a rate of about 1.14% per year. To find the time when this population is likely to double, you need to solve the equation 6.2(1.0114)t = 12.4, or (1.0114)t = 2 As you work on the problems of this investigation, look for ways to answer this question:

How can common logarithms help in finding solutions of exponential equations? 1) Use number sense and what you already know about logarithms to solve these equations.

a. 10x = 1,000 b. 10x + 2 = 1,000 c. 103x + 2 = 1,000 d. 2(10)x = 200 e. 3(10)x + 4 = 3,000 f. 102x = 50 g. 103x + 2 = 43 h. 12(10)3x + 2 = 120 i. 3(10)x + 4 + 7 = 28

Unfortunately, many of the function that you have used to model exponential growth and decay have not used 10 as the base. On the other hand, it is not too hard to transform any exponential expression in the form bx into an equivalent expression with base 10. You will learn how to do this after future work with logarithms. The next three problems ask you to use what you already know about solving exponential equations with base 10 to solve several exponential growth problems.

2) If a scientist counts 50 bacteria in an experimental culture and observes that one hour later the count is up to 100 bacteria, the function P(t) = 50(100.3t) provides an exponential growth model that matches these data points. a. Explain how you can sure that P(0) = 50. b. Show that P(1) 100.

c. Use the given function to estimate the time when the bacteria population would be expected to

reach 1,000,000. Explain how to find this time in two ways – one by numerical or graphic estimation and the other by use of logarithms and algebraic reasoning.

3) The world population in 2005 was 6.2 billion and growing exponentially at a rate of 1.14% per year. The function P(t) = 6.2(100.005t) provides a good model for the population growth pattern.

a. Explain how you can be sure that P(0) = 6.2

b. Show that P(1) = 6.2 + 1.14%(6.2)

c. Find the time when world population would be expected to reach 10 billion if growth

continues at the same exponential rate. Explain how to find this time in two ways – one by numerical or graphic estimation and the other by use of logarithms and algebraic reasoning.

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a. ________________________________________________________________________ b. ________________________________________________________________________ c. ________________________________________________________________________

Use logarithms and other algebraic methods as needed to complete the following tasks.

a. Solve these equations. I. 5(10)x = 450

II. 4(10)2x = 40 III. 5(10)4x – 2 = 500 IV. 8x2 + 3 = 35

b. The population of the United States in 2006 was about 300 million and growing exponentially at a rate of about 0.7% per year. If that growth rate continues, the population of the country in year 2006 + t will be given by the function P(t) = 300(100.003t). According to that population model, when is the U.S. population predicted to reach 400 million? Check the reasonableness of your answer with a table or a graph of P(t).

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Independent Practice with Common Log Word Problems 1) Use algebraic reasoning with logarithms to solve the following equations for x.

a. log x = 2 b. 15 = 10x c. 5(10)2x = 60 d. 103x – 1 = 100,000

2) When archaeologists discover remains of an ancient civilization, they use a technique called carbon dating to estimate the time when the person or animal died or when the artifact was made from living material.

The amount of radioactive Carbon-14 in such an artifact decreases exponentially according to the function C(t) = 100(10-0.00005255t), where t is time in years and C(t) is the percent of the Carbon 14 present in the artifact when it was last living material.

a. What is the half-life of Carbon-14, the time when only 50% remains from an original amount?

b. Suppose that a skeleton is discovered that has only 10% of the Carbon-14 that one finds in living animals. When was that skeleton part of a living animal?

3) The Washington Nationals baseball team was purchased in 2006 for 450 million dollars. If the value

of this investment grows at a rate of 5% compounded yearly, the purchase price of the team in 2006 + t will be given by V(t) = 450(100.021t).

a. Explain how you can be sure that this function gives the correct value of the investment in 2006.

b. Use the function to estimate the value of the investment in 2010.

c. Use logarithms and other algebraic reasoning to estimate the time when the value of the investment

will be $1 billion ($1,000 million).

4) Suppose that the average rent for a two-bedroom apartment in Indianapolis is currently $750 per

month and increasing at a rate of 8% per year. The function R(t) = 750(100.033t) provides a model for the pattern of expected increase in monthly rent after t more years. a. Explain how you know that R(0) = 750.

b. Find the time when the average rent for a two-bedroom apartment will be $1,000 per month, if inflation continues at the current 8% rate. Show how to find that time in two ways – one by estimation using a table or graph of R(t) and another using logarithms and algebraic reasoning.

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5) If an athlete tries to improve his or her performance by taking an illegal drug, the amount of that drug

in his or her blood will decline exponentially over time, but tests are quite sensitive to small amounts.

For example, a 200 mg dose of a steroid might decay so that the amount remaining after t days is given by the rule s(t) = 200(10-0.046t). a. Explain how you know that s(0) = 200.

b. Estimate the time when only 5 mg of the steroid remains active in the athlete’s blood. Show how to find that time in two ways – one by estimation using a table or graph of s(t) and another using logarithms and algebraic reasoning.

6) Suppose that 500 mg of a medicine enters a hospital patient’s bloodstream at noon and decays

exponentially at a rate of 15% per hour. a. Use the exponential function D(t) = 500(10-0.07t) to predict the amount of medicine active in the

patient’s blood at a time 5 hours later, where t is time in hours.

b. Find the time when only 5% (25 mg) of the original amount of medicine will be active in the patient’s body. Show how to find that time in two ways – one by estimation by using a table or graph of D(t) and another using logarithms and algebraic reasoning.

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Common Core Math II Name Date

Exponential Functions Review 7) Use the rules of exponents to find the value of x in each equation.

a. (2x+1)5 = 2x b. (32x)(316) = 348 c. (550)/(540) = 5x-5

8) Use your calculator to find the following logarithms.

a. log -100 b. log 426 c. log 100 d. log (0.0001) e. log 3.45

9) Use your knowledge of exponents and logarithms to solve these equations two ways. a. 3(10x) = 3,000 b. 102x-1 = 100 c. 102x - 3 = 997 d. 10x = 1

e. -2(10)x + 4 = -.002 f. 10x/2 = 25 g. (10)x + 2 = 1500 h. 3(10)x + 4 + 3 = 15

10) A 100 milligram sample of Carbon-10 has a half-life of 19.29 seconds. Write an exponential function

to model its decay. Let x= time in minutes and f(x) = the amount of Carbon-10 remaining in the sample.

11) Create a real world scenario that could be modeled by the function . In your

scenario, make sure to address percentage of exponential growth or decay and initial value.

12) A popular antique is gaining value because it is so hard to find. In 1985 its value was $125, and in

2000 its value was $1925.90. a. Find an explicit exponential function to model the information – show your work.

b. Write a recursive (NOW-NEXT) function to model the data.

c. Determine the percentage of yearly appreciation.

d. If the same trend continues, how much was the antique worth in 2010?

Use what you know about solving exponential equations with base 10 to solve the following growth problem.

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13) In a drop of pond water, there are 18 protozoa. Ten hours later, there are 180 protozoa in the dish. P(t) = 18(100.1t) provides an exponential growth model that matches these data points. a. Verify that the model P(t) = 18(100.1x) represents the information provided.

b. Use the given function to estimate the time when the bacteria population would be expected to

reach 500,000. i. Explain how to find the time by numerical or graph estimation.

ii. Explain how to find the time by using common logarithms and algebraic reasoning.

c. What is the theoretical domain of the function?

d. What is the practical domain of the function?

e. What is the range of the function?

f. What are the intercepts of the function and what do they mean in the context of the problem?

g. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem?

14) For the function f(x) = (0.75)x - 1 evaluate the following: a. f(-1) = b. f(0) = c. f(5) = d. f(2) =

22)

Describe the effect on the graph when…

y = a ∙ bx+c + d y = a ∙ log (bx + c) + d

a is negative

a increases

b increases

c decreases

d increases

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23) For each of the functions describe the key characteristics.

y = -2x+4 - 3 y = log(x+2)

Domain

Range

Asymptotes (if any)

Zeros (if any)

End behavior as x

End behavior as x -

Sketch of the function

25) Given the function , answer the following questions.

a. What is the inverse of the function? b. How can you verify algebraically that the functions are inverses? c. How can you verify graphically that the functions are inverses?