Objective 21 Inverse Functions

Illustrate the idea of inverse functions.

f(x) = x2+1

x

y

f(x) =√x− 1

x

y

—————————————————————————————————————–

Two one-to-one functions are inverses of each other if (f ◦ g)(x) = for all x in the domainof g, and (g ◦ f)(x) = for all x in the domain of f .

We write f−1 to denote the inverse function.

Objective 21b How are the graphs of f and f−1 related?

x

y

If (a, b) is on the graph of y = f(x), then is on the graph of y = f−1(x).

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Objective 21b Example Select the graph of y = f−1(x).

x

y

x

y

x

y

x

y

x

y

A function can be its own inverse. Consider

x

y

Objective 21a Does every function have an inverse?

x

y

x

y

Graph of a function must pass the to be be the graph of a one-to-one function.

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Which are one-to-one functions?

{(1, 2), (1, 3), (5, 4)} {(1, 2), (3, 2), (4, 5)} {(1, 2), (3, 3), (4, 5)}

If a function is not one-to-one, restrict the domain in order to define an inverse function. (Recallintro to Obj 21.)

x

y

Objective 21d Given a function, find the function rule for f−1.

***************Plan of Attack 1. Write y for f(x) (to simplify the notation).

2. Solve for x. For applied mathematicians, when units are usually associated with the variables,you have the inverse function.

3. For our College Algebra course, we will interchange x and y to write the inverse as a function ofx.

4. Write f−1(x) for y (to return to function notation).

***************

Find the function rule for f−1 for each of the following.f(x) = 3x− 5 f(x) = (4− x)5 + 7

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f(x) = 3√x− 2 + 7 f(x) =

x+ 1

x− 3

f(x) =2x− 3

5− x

Objective 22 Exponential Functions. f(x) = ax, a > 0, a 6= 1

Does this define a function?

Don’t allow base to be negative because could be for some x. i.e.

Don’t allow base to be 1 because , graph would be linear, not exponential.

What’s the domain? “All reals?” If so, we have to define what’s meant by irrational exponents.

For example: 4√2 or 4π We haven’t worked with irrational exponents.

Good News: The limiting processes of calculus guarantee that irrational exponents are defined, and“line up” as we want.

That means, since 3 < π < 4, then a3 < aπ < a4.

That means, since 1 <√2 < 2, then a1 < a

√2 < a2.

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The exponential functions are classified into 2 groups, depending on the base.

f(x) = ax, a > 1 f(x) = ax, 0 < a < 1

We will consider two specific cases to develop the concept. This is not an eGrade example; you willnot be making tables of values - you will not be plotting points.

Consider f(x) = 4x Consider f(x) =(

1

4

)x

for an example of a > 1 for an example of 0 < a < 1

x y x y x y x y

−1000 1 −1000 1

−100√2 −100

√2

−10 2 −10 2

−3 3 −3 3

−2 10 −2 10

−1 100 −2 100

0 1000 0 1000

x

y

x

y

Objective 22a Properties and Graphs of Exponential Functions

f(x) = ax, a > 1 f(x) = ax, 0 < a < 1

x

y

x

y

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Objective 22b Graphing Exponential Functions with Reflections or Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = −5x f(x) =(

1

4

)−x

y

x

y

x

1

y

x

1

y

x

Which function best describes the graph shown? Which function best describes the graph shown?y

x

1

y

x

f(x) = (2.5)−x f(x) = −(2.5)x f(x) = (2.5)−x f(x) = −(2.5)x

f(x) = (0.4)−x f(x) = −(0.4)x f(x) = (0.4)−x f(x) = −(0.4)x

More Objective 22b Graphing Exponential Functions with Translations

Don’t . Don’t . Use Obj 14!

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Select the graph that best represents the graph of each of the following.

f(x) = 4x − 3 f(x) =(

15

)x+2

y

x

y

x

y

x1

y

x

1

y

x

y

x

1

y

x

y

x

1

Which function best describes the graph shown? Which function best describes the graph shown?y

x

1

y

x

f(x) = 6x+3 f(x) = 6x + 3 f(x) =(

52

)x+2f(x) =

(

52

)x−2

f(x) = (0.6)x+3 f(x) = (0.6)x + 3 f(x) =(

25

)x−2f(x) =

(

25

)x+2

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Objective 22c “The” exponential function is f(x) = ex because of so many areas of application.

e ≈ 2.71828 e = limn→∞

(

1 +1

n

)n

Graph f(x) = ex

Evaluate ex on a scientific calculator (the Mac calculator in lab class).

Strontium 90 is a radioactive material that decays over time. The amount, A, in grams of Strontium90 remaining in a certain sample can be approximated with the function A(t) = 225e−0.037t , wheret is the number of years from now. How many grams of Strontium 90 will be remaining in thissample after 7 years?

$8,000 is invested in a bond trust that earns 5.9% interest compounded continuously. The accountbalance t years later can be found with the function A = 8000e0.059t. How much money will be inthe account after 6 years?

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Objective 22d Solving exponential equations when we can obtain the same base.

Exponential functions are one-to-one; that means:

if and only if

Rewrite each side (if needed) in terms of a common base; use the smallest base possible. Be sureto replace equals.

Solve 52x+1 = 253−x Solve(

4

9

)x−4

=(

27

8

)3x

Solve(

1125

)4x−1= 57x+5

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Objective 23 Logarithmic Functions

Consider an exponential function y = ax

What’s the inverse function?

There is no algebraic operation to solve for x.

We must define a new function. y = loga x

Objective 23a Evaluate Logarithmic Functions

log2 8 =

log25 5 =

log1/16 2 =

log2 2 =

log2 1 =

Which are defined? (Be careful, sometimes ask “Which are undefined?”)

log1/2 1 log1/4 4 log1/2(−4) log1/2 0

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Objective 23b Properties and Graphs of Logarithmic Functions f(x) = logb x, b > 0, b 6= 1

The logarithmic functions are classified into two groups comparable to the exponential functions.

Recall Obj 22a

x

y

x

y

y = ax, a > 1 y = ax, 0 < a < 1

x

y

x

y

y = logb x, b > 1 y = logb x, 0 < b < 1

Consider f(x) = log4 x Consider f(x) = log1/4 xfor an example of b > 1 for an example of 0 < b < 1

Pick the y-values, find the x-values. Pick the y-values, find the x-values.x y x y

−2 −2

−1 −1

0 0

1 1

2 2

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Objective 23c Graphing Logarithmic Functions with Reflections or Translations

Don’t . Don’t . Use Obj 14!

Select the graph that best represents the graph of each of the following.

f(x) = − log4 x f(x) = log1/4(−x)

1

y

x 1

y

x 1

y

x 1

y

x

Which function best describes the graph shown? Which function best describes the graph shown?

1

y

x 1

y

x

f(x) = − log(5/2)(x) f(x) = log(5/2)(−x) f(x) = − log(5/2)(x) f(x) = log(5/2)(−x)

f(x) = − log(2/5)(x) f(x) = log(2/5)(−x) f(x) = − log(2/5)(x) f(x) = log(2/5)(−x)

More Objective 23c Graphing Logarithmic Functions with Translations

Don’t . Don’t . Use Obj 14!

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Select the graph that best represents the graph of each of the following.

f(x) = log3(x) + 2 f(x) = log1/3(x+ 2)

1

y

x

1

y

x

y

x

y

x

1

y

x

1

y

x

y

x

y

x

Which function best describes the graph shown? Which function best describes the graph shown?

1

y

x

y

x

f(x) = log(5/2)(x) + 2 f(x) = log(5/2)(x)− 2 f(x) = log(5/2)(x+ 2) f(x) = log(5/2)(x− 2)

f(x) = log(2/5)(x) + 2 f(x) = log(2/5)(x)− 2 f(x) = log(2/5)(x+ 2) f(x) = log(2/5)(x− 2)

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Objective 23d Domain of Logarithmic Functions (not by graphing)Give the domain.f(x) = logb(4− 5x)

f(x) = 15− logb(3x)

f(x) = logb

(

x+ 1

x− 3

)

f(x) = log3(4− x2)

f(x) = log3(x2 + 4)

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Objective 24 Properties of Logarithmic Functions

As used below: a > 0, a 6= 1, b > 0, b 6= 1, M > 0, N > 0, x > 0, y and r represent any real number

Definition - Obj 24a means

Common Logarithms are logarithms base 10; we write instead of .

Natural Logarithms are logarithms base e; we write instead of .

Objective 24a Example Which of the following is equivalent to ln 5 = x?

A) 5e = x B) ex = 5 C) x5 = e

Properties of Logarithms - Obj 24b

Product Rule logb(MN) =

Must Note: logb(MN) 6=

Must Note: logb(M +N) 6=

Quotient Rule logb

(

M

N

)

=

Must Note: logb

(

M

N

)

6=

Must Note: logb(M −N) 6=

Power Rule logb Mr =

Change-of-Base Formula logb M = logb M =

When Base and Result Match logb b =

When Result is 1 logb 1 =

91

Inverse Function Properties - Obj 24c Recall Obj 21: (f ◦ f−1)(x) = x and (f−1 ◦ f)(x) = x

aloga M = loga ar =

Objective 24c Examples

Solve for x if 5log5(3x) = 15 Solve for x if ln e15x = 3

Objective 24b Example Which of the following is equivalent to logb(x− y)?

A) logb

(

x

y

)

C) both A and B

B) logb x− logb y D) none is equivalent

Objective 24e Examples Evaluating ex or ln x on a scientific calculator and rounding the resultto 3 decimal places.

When rounding a number to 3 decimal places, look at the 4th digit of the decimal.

If that number is less than 5, then keep the 3rd digit as it is and drop the remaining decimal digits.

If that number is greater than, or equal to 5, then round the 3rd digit up (add 1 to the 3rd digit)and drop the remaining decimal digits.

Evaluate, round to 3 decimal places. e−5 ≈ ln 50 ≈

Objective 24be Example Select all that are correct for log3 8.

Choice: =log 8

log 3Choice: =

ln 3

ln 8Choice: ≈ 1.983 Choice: ≈ 0.528

Objective 24abc Example Select ALL the correct formulas/statements if b > 0, b 6= 1, x > 0, y >

0. (eGrade does not warn when a multiple selection problem is left blank.)log(x+ y) = log x+ log y

log(x− y) = log x− log y

log x = y means 10x = y

log5 1 = 0

If log 10−4x = −12, then x = 3

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Applying Log Properties - Objective 24d

Expand using log properties. logb

(

x2

yz

)

Expand using log properties. logb

(

x2y

z(w + 3)

)

Which of the following is equivalent tologb(x

2y)

logb(z(w + 3))

A)2 logb x+ logb y

logb z + logb(w + 3)

B) 2logbx+ logb y − logb z − logb(w + 3)

C) A and B are the same

another Objective 24d Example

Write as a single logarithm 2 logb x− logb y +12logb z

A) logbx2

y√z

B) logbx2√z

y

93

another Objective 24d Example

Write as a single logarithm. 2 logb(z − w)− logb w + 3 logb z + logb x− logb(x+ w)

another Objective 24d Example

If logb 2 = l and logb 5 = m, express logb 100 in terms of l and m.

another Objective 24d Example

If logb 2 = l and logb 5 = m, express (logb 4) · (logb 25) in terms of l and m.

Objective 25 Solving Exponential Equations when we can’t obtain the same base.

Logarithmic functions are one-to-one. That means: if and only if

Objective 25a

Solve. 62x−1 = 143−x Solve. 25x+3 = 12

94

Objective 25b Round solution to 3 decimal places.Solve. 1400e3x+1 = 42000 Solve. 40e2x = 120

Solve. 1400 · 103x+1 = 42000 Solve. 40 · 102x = 120

Objective 26 Solving Logarithmic Equations

Objective 26a All terms in the equation involve log functions.

Solve. log x+ log(x− 2) = log(x+4) Solve. log3(11 + x)− log3(x+7) = log3(x+5)

95

Solve. log7(x−8)+log7(x−9) = log7(23−3x) Solve. log5(x+4) = log5(3x+8)−log5(x+3)

Objective 26b Use the Definition of logarithmic notation.

Solve. 5− 7 log1/3 x = 5 Solve. log(x2 + 2x+ 11) = 1

Solve. 6 log4(x− 5) = 3 Solve. log2(x2 + 7) = 4

96

Objective 27 Solving Equations that involve exponential functions.

Solve.e2x · 4x3 − x4 · e2x · 2

e4x= 0 Solve.

e3x · (−x3) + x · e3x · 4e6x

= 0

Solve.(−x2) · ex − ex · 4

e2x= 0 Solve.

2e2x(4x+ 5)3 − e2x · 3 · (4x+ 5)2 · 4(4x+ 5)6

= 0

Solve.(3x+ 7)4 · e4x · 4 − e4x · 4(3x+ 7)3 · 3

(3x+ 7)8= 0

97

Solve. Solve.(x+ 2)6 · e3x · 2 − 5(x+ 2)4 · 2 · e3x

(x+ 2)8= 0

2 · (x− 3)5 · e4x · 2 − e4x(x− 3)3 · 4(x− 3)6

= 0

Objective 28 Solving Equations that involve natural logarithmic functions.

Solve. 22−(

12 lnx+ 12x ·(

1

x

))

− 7 = 0 Solve. 44−(

8x(

1

x

)

+ 8 lnx)

= 0

Solve.

(

2

x

)

· x− 2 ln x

x2= 0 Solve.

x4 · 3x

− (3 lnx)(4x3)

x8= 0

98

Solve.4x3 · ln x ·

(

1

x

)

− 16 lnx

(lnx)2= 0 Solve.

x3 ln x − 3x4 · ln x(

3

x3

)

x6= 0

Objective 30 Solve 2 linear equations in 2 unknowns

There are 3 possible situations.

x

y

x

y

x

y

To solve algebraically we will use the

Multiply one, or both equation if needed, by non-zero numbers so that when the equations areadded, one variable is eliminated.

99

Solve. 2x+ 5y = 10 Solve. 7x− 4y = −143x− 2y = 6 3x− 2y = 4

Solve. 9x− 12y = 9 Solve. −6x+ 10y = −421x− 28y = 21 21x− 35y = −14

Objective 31 Solve 2 equations in 2 unknowns

First type: Quadratic and Linear - Algebraic solution

To solve algebraically we will use the

Substitute the into the .

Solve. y = x2 + 2 Solve. y = x2 − 412x− 4y = 1 5x− 3y = 15

Give the x-coordinate(s) only of any solution(s). If multiple solutions, then enter the values in anyorder, separated by a semicolon. If there is no solution, then enter: no solution (When there is nosolution, do not use any capital letters; do not use any punctuation marks.)

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Second type: 2 Various equations - Solve by GRAPHING

We will ask: “ solutions are there?”

Must know all the basic functions and equations we have graphed and

we must know Obj 14 - and

x

y

x

y

x

y

x

y

x

y

x

y

x

y

1

y

x

1

y

x

x

y

( h,k)

( x,y)

r

101

Solve the system of equations by graphing. How many real solutions are there?y =

√x+ 1

y = |x| − 2

Solve the system of equations by graphing. How many real solutions are there?y = −ex

y = 3√x− 1

Solve the system of equations by graphing. How many real solutions are there?y = ln x(x− 1)2 + y2 = 4

Copyright c©2010-present, Annette Blackwelder, all rights are reserved. Reproduction or distribu-tion of these notes is strictly forbidden.

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