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Chapter 12 Handouts Page 1 of 41

Page 1 of 41

Section 12.1 – Exponential Functions

Objectives:

Graph an exponential function.

Solve applied problems involving exponential functions.

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Page 2 of 41

Example: Suppose you have a large sheet of paper 0.015 cm thick, and you tear the

paper in half and put the pieces on top of each other. Continue tearing and stacking the

paper in this manner, always tearing each piece in half. How high would the resulting

pile of paper be if you continued the process of tearing and stacking 20 times?

No. of tears Height of stack (cm)

0 0.015


Page 3 of 41

Definition of Exponential Functions

What makes an exponential function different from a polynomial function is that the

independent variable (x) is .

Graphing Exponential Functions y = f(x)

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Plot: f(x) = 2x

x f(x)

–3

–2

–1

0

1

2

3

Plot: g(x) =

x

2

1

x f(x)

–3

–2

–1

0

1

2

3


Page 4 of 41

Graphing Exponential Functions x = f(y)

This time, x is where the y usually is (on the left-hand side), and y is up in the exponent.

Do the reverse of what we usually do: pick values, and use them to calculate the

values. Then, plot the ordered pairs as usual.

Example: x = 2y x y -3 -2 -1 0 1 2 3

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Page 5 of 41

1. Graph: f(x) = 3x+2

x y

-4

-3

-2

-1

0

1

2. Graph: x = (2

3)

𝑦

Note: just approximate the x values with decimals for plotting.

x y

-4

-3

-2

-1

0

1

2

3

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Page 6 of 41

Applications of Exponential Functions

where:

P = amount ($) of

r = interest rate in

n = number of times interest is compounded per year

t = time period,

A = total amount ($) after

Example: Suppose that $10,000 is invested at 3.2% interest, compounded semi-

annually. Find a function A for the amount in the account after t years. Complete the

following table of function values.

List what we know:

P =

r =

n =

Compound interest formula: 𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡

t A(t)

0

1

2

5

10

20


Page 7 of 41

1. The Stephensons invest $4000 in an account paying 2.8%, compounded

quarterly. Find the amount in the account after 3 ½ years.

2. Knee replacement in the United States for people ages 45 – 64 has increased

exponentially since 2000. The number of knee replacements K(t) performed t

years after 2000 can be approximated by:

K(t) = 90,892(1.12)t

where t = 0 corresponds to 2000.

How many knee replacements were performed for people ages 45-64 in 2009?


Page 8 of 41

Section 12.2 – Inverse Functions and Composite Functions Objectives:

Determine if a function is one-to-one.

Find the inverse of a one-to-one function.

Graph a function and its inverse.

Find the composition of two functions.

Definition of an inverse relation:

In other words, swap the of each ordered pair.

Example: f = { (0, 1), (1, 3), (2, -2), (3, 5) }

Inverse relation is:

Plot of f and inverse relation:


Page 9 of 41

One-to-One Functions

Goal: find the inverse of a function. The inverse of a function is another function.

For a function f: as in f(x)

Its inverse function is called

HOWEVER, a function f(x) only has an inverse if it is a function.

One-to-one means:

Each x has a , or

Can’t have more than one x going to the .

Important Results:

If f is not one-to-one

f does have an inverse

f-1 does !

One-to-One Function: NOT One-to-One Function: NOT A FUNCTION!


Page 10 of 41

Determining if a Function is One-to-One

Example: f(x) = x3 – 2x2 + 2


Page 11 of 41

Sketch the following types of functions:

Linear (not horizontal or vertical): Quadratic:

One-to-one? One-to-one?

Absolute value: Square Root:

One-to-one? One-to-one?

Exponential:

One-to-one?


Page 12 of 41

Assuming that a function is one-to-one, there are two things we are going to do:

Sketch the graph of the inverse function.

Find the equation of the inverse function.

To sketch the graph of the inverse of a function, just the ordered

pairs of the function.

Example: The graph of a one-to-one function f(x) is given. Draw the graph of the

inverse function f -1.

Ordered pairs in f(x):

Ordered pairs in f-1(x):

- 5

- 4

- 3

- 2

- 1

0

1

2

3

4

5

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5


Page 13 of 41

Find the Inverse of a One-to-One Function

Example: f(x) = 2

1x – 3 Find the inverse of f(x).

1. Rewrite as:

2. Exchange x and y:

That IS the inverse! BUT, normally we write equations as y = f(x), not x = f(y), so then

we just

3. Solve the equation for y:

Add 3 to both sides:

Multiply by 2:

4. Write in inverse notation:

Plot f(x) and its inverse: (next page)


Page 14 of 41

f(x) = 2

1x – 3

f-1(x) =

Example: f(x) = 2𝑥−1

5𝑥+3 Find the inverse.

1. Rewrite as:

2. Exchange x and y:

3. Solve the equation for y:


Page 15 of 41

Summary of Important Stuff about Inverse Functions

1. Inverse functions undo each other, so they reverse the ordered pairs: (x, y)

becomes .

2. Only functions have an inverse!

3. Horizontal line test will determine if a function has an inverse.

4. The graphs of a function and its inverse are symmetric with respect to the line

.

The following functions are one-to-one. Find the inverse function for each.

1. f(x) = −7

8𝑥 + 2

CONTINUED ON NEXT PAGE!


Page 16 of 41

2. f(x) = √𝑥 − 43

3. 𝑓(𝑥) =1

𝑥−8


Page 17 of 41

Composite Functions

Definition: the composition of two functions means to combine two functions by

substituting one function’s formula in place of each in the other function’s

formula.

Given two functions:

The composition of functions f and g is:

Defined/written as:

gf (x) or ( gf )(x)

gf means:

use the expression for g(x) as the

The composition of functions g and f is:

Defined/written as:

fg (x) or ( fg )(x)

fg means:

use the expression for f(x) as the

ORDER is VERY IMPORTANT, because composition of functions is not commutative!!

In other words:


Page 18 of 41

Example: f(x) = 4x + 3 g(x) = 2x2 – 5

gf (x): fg (x):

Example: f(x) = 2

1

x g(x) = x – 1

gf (x): fg (x):

Practice: f(x) = 2 – x g(x) = 2x2 + 1

Find:

gf (x): fg (x):


Page 19 of 41

Section 12.3 – Logarithmic Functions Objectives:

Graph logarithmic functions.

Convert from exponential equations to logarithmic equations and vice versa.

Solve logarithmic equations.

Find common logarithms on a calculator.

Example Exponential Function: y = 2x

Find the inverse analytically:

1. Write the function:

2. Swap x and y:

3. Solve for y:

Describe the inverse: y =

Introduce a new notation: called

For: x = 2y define:

Notice that the “base 2” appears in both functions.

Recap:

Start with: y = 2x Inverse: (written as x in terms of y)

(written as y in terms of x)

These inverse functions are

BOTH describe the same graph – the one of f -1 that you plotted.

Logarithmic function is the of the exponential function.

A is an .


Page 20 of 41

f(x) = 2x f -1(x) = Domain of f: Domain of f -1:

Range of f: Range of f -1:

x y = 2x

Find f -1 by reversing

ordered pairs:

x y

-3

-2

-1

0

0.5

1

1.5

2

* do all calculations to three decimal places

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

Instructions:

For the function: f(x) = 2x 1. In the first table (bottom left), calculate y for the given values of x. (Round answers off to 3 decimal places). 2. Plot the function, and list domain and range. 3. Find the inverse function by reversing the ordered pairs. Fill out the data table for f -1(x). 4. Plot the inverse function.


Page 21 of 41

Example: Graph y = f(x) = log3x

Start by converting the logarithmic equation to an exponential equation.

Calculate x for the given values of y, and plot the function.

x y

-3

-2

-1

0

1

2

3


Page 22 of 41

Convert Between Exponential and Logarithmic Equations

Examples: 2 = log7 49

9 = 811/2

Problems:

1. Convert to an exponential equation: t = log59

2. Convert to a logarithmic equation: 161/4 = 2

Two more properties of logarithms:


Page 23 of 41

Solving Certain Logarithmic Equations

Example: Solve: logx49 = 2

Technique: convert to an exponential equation.

Note: all logarithm bases must be .

Example: Find log819

Technique: set the logarithm equal to x:

Convert to an exponential equation:

Problems:

1. Solve: log4 x = 3

2. Solve: logx 1000 = 3

3. Find: log8 64

4. Find: 𝑙𝑜𝑔51

125


Page 24 of 41

Common Logarithms

The “common logarithm” is defined as log base .

Therefore: log a =

Graph of the function: f(x) = log x

From the graph: log 10 =

For other values, just use the ‘log’ button on your calculator.

Example: log 20 =

Find the common logarithm to four decimal places:

1. log 0.25 =

2. log 5486 =

3. log -92 =


Page 25 of 41

Section 12.4 – Properties of Logarithmic Functions Objectives:

Learn properties of logarithms, and how to use them to manipulate logarithmic

expressions.

The Product Rule

Example: log2 (84)

Example: loga 75 + loga 5

The Power Rule

Example: log3 x2

Example: 𝑙𝑜𝑔9√7


Page 26 of 41

The Quotient Rule

Example: 𝑙𝑜𝑔𝑎14

𝑥

Example: 𝑙𝑜𝑔𝑏21 − 𝑙𝑜𝑔𝑏3

Logarithm of a Base to a Power

Remember that logarithms and exponentials are functions, so they

each other.

Example: log 105.6


Page 27 of 41

Combining multiple rules:

Express in terms of logarithms of a single variable or a number.

Example: loga5xy4z3

Example: 𝑙𝑜𝑔𝑎√𝑎6𝑏8

𝑎2𝑏5

Example: Express as a single logarithm and, if possible, simplify.

𝑙𝑜𝑔𝑎(𝑥2 − 4) − 𝑙𝑜𝑔𝑎(𝑥 − 2)


Page 28 of 41

Problems:

Express as a single logarithm:

1. logd54 – logd9

Express in terms of logarithms of a single variable or a number:

2. 𝑙𝑜𝑔𝑏𝑝2𝑞5

𝑚4𝑛7

3. 𝑙𝑜𝑔𝑎𝑥2 − 2𝑙𝑜𝑔𝑎√𝑥


Page 29 of 41

Section 12.5 Natural Logarithmic Functions

Objectives:

Define the number ‘e’, and exponents or logarithms base e

Use the change-of-base formula

Graph exponential and logarithmic functions base e

Exponential Functions with Base e

General exponential function: vs. Exponential function with base e:

What is e?

It’s an number, like .

It equals . Find your ‘e’ button on calculator.

It is the base amount of growth shared by all continually growing processes.

Example of growth:

Compounded how often? Amount at end of one year

once (end of year)

twice (mid/end of year)

quarterly (4 times)

monthly (12 times)

daily (365 times)

every hour (8760 times)


Page 30 of 41

THE Exponential Function: y = ex

Find the inverse analytically:

1. Write the function:

2. Swap x and y: this is the inverse!

3. Solve for y*: this is also the inverse!

*In this case, we already know how to solve for the inverse!

If y = ax, then the inverse function is: y =

In this case, since y = ex, the inverse is: Notation for logex:

Called: the logarithm

Written:

Find it on your calculator! ln 5 = ln 0.04 =


Page 31 of 41

f(x) = ex f -1(x) = Domain of f: Domain of f -1:

Range of f: Range of f -1:

x y = ex

Find f -1 by reversing

ordered pairs:

x y

-3

-2

-1

0

0.5

1

1.5

* do all calculations to three decimal places

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

Instructions:

For the function: f(x) = ex 1. In the first table (bottom left), calculate y for the given values of x. (Round answers off to 3 decimal places). 2. Plot the function, and list domain and range. 3. Find the inverse function by reversing the ordered pairs. Fill out the data table for f -1(x). 4. Plot the inverse function.


Page 32 of 41

Graph. 1. f(x) = e–x + 3

x y

-1.5

-1

0

1

2

3

4


Page 33 of 41

Graph. 2. f(x) = 2ln(x – 2)

x y

2.1

2.5

3

4

5

6

7


Page 34 of 41

Change of Base Formula

To calculate logbM for bases ‘b’ other than 10 or ‘e’:

Example: log3100

Calculate using log10:

OR

Calculate using loge:

Example: log542

Calculate using log10:

OR

Calculate using loge:


Page 35 of 41

Section 12.6 Solving Exponential Equations and Logarithmic Equations

Objectives:

Solve exponential equations.

Solve logarithmic equations.

Solving exponential equations

Technique: write both sides of the equation as powers of the same .

Then, equate the .

Example: 32x = 9

Example: 42x-3 = 64


Page 36 of 41

Using the Principle of Logarithmic Equality

If you cannot write both sides of the equation with the same base, then a different

technique is used:

If the equation is: x = y In other words:

THEN, take the of both sides of the equation.

Why? Because then we can apply the “Power Rule” to get the exponent “out”.

Example: 7x = 20

Take log of both sides:

Apply power rule:

Solve:


Page 37 of 41

Practice – solving exponential equations

1. 57x = 625 hint: rewrite 625 as a power of 5

2. 6x = 10


Page 38 of 41

Solving Logarithmic Equations

Technique: convert the logarithmic equation to an equivalent exponential equation.

How to remember this? Think in terms of inverse functions. Logs and exponents are

inverse functions, so they “undo” each other.

Example: y = logax Inverse of loga is a^… so do a^ on both sides.

Example: ay = x Inverse of ay is loga… so take loga of both sides


Page 39 of 41

Example: log5x = 2

Rewrite as an exponential equation:

(Exponentiate both sides with a base of 5)

Example: log3(5x + 7) = 2

Rewrite as an exponential equation:

(Exponentiate both sides with a base of 3)

Using additional properties to solve logarithmic equations

You may have to do an additional step in advance, which is to combine logarithmic

expressions into a SINGLE logarithmic expression:

from Section 12.4


Page 40 of 41

Example: log x + log(x + 3) = 1

First step: combine the log expressions on the LHS using Property 1:

Second step: convert to an exponential equation:

Example: log3(2x – 1) – log3(x – 4) = 2

First step: combine the log expressions on the LHS using Property 3:

Second step: convert to an exponential equation:


Page 41 of 41

Solve.

1. 𝑙𝑜𝑔9𝑥 =1

2

2. log2(8 – 2x) = 6

3. log x + log(x + 9) = 1

4. log4(x + 3) – log4(x – 5) = 2

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