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Chapter 5: Exponential and Logarithmic Functions

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5.1 Inverse Functions

5.2 Exponential Functions

5.3 Logarithms and Their Properties

5.4 Logarithmic Functions

5.5 Exponential and Logarithmic Equations and Inequalities

5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

Example

Also, f[g(12)] = 12. For these functions, it can be

shown that

for any value of x. These functions are inverse functions

of each other.

A function f is a one-to-onefunction if, for elements a and b from the domain of f,

a b implies f(a) f(b).

- Only functions that are one-to-one have inverses.

Example Decide whether the function is one-to-one.

(a) (b)

Solution

(a) For this function, two different x-values produce two different y-values.

(b)If we choose a = 3 and b = –3, then 3 –3, but

If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.

Example Use the horizontal line test to determine

whether the graphs are graphs of one-to-one functions.

(a) (b)

One-to-one

Not one-to-one

Let f be a one-to-one function. Then, g is the inversefunction of f and f is the inverse of g if

Example

are inverse functions of each other.

- Notation for the inverse function f-1 is read
“f-inverse”

Finding the Equation of the Inverse of y = f(x)

1. Interchange x and y.

2. Solve for y.

3. Replace y with f-1(x).

Any restrictions on x and y should be considered.

ExampleFind the inverse, if it exists, of

Solution

Write f(x) = y.

Interchange x and y.

Solve for y.

Replace y with f-1(x).

- f and f-1(x) are inverse functions, and f(a) = b for real numbers a and b. Then f-1(b) = a.
- If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f-1.

If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x.

ExampleLet

SolutionNotice that the domain of f is restricted

to [–5,), and its range is [0, ). It is one-to-one and

thus has an inverse.

The range of f is the domain of f-1, so its inverse is

- If f is one-to-one, then f-1 exists.
- The domain of f is the range of f-1, and the range of f is the domain of f-1.
- If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f-1, so the graphs of f and f-1 are reflections of each other across the line y = x.

Example Use the one-to-one function f(x) = 3x + 1 and the

numerical values in the table to code the message BE VERY CAREFUL.

A1F6K 11P 16U21

B 2G 7L 12Q 17V22

C 3H 8M 13R 18W23

D4I 9N 14S 19X24

E 5J 10O 15 T 20Y 25

Z 26

SolutionBE VERY CAREFUL would be encoded as

7 16 67 16 55 76 10 4 55 16 19 64 37

because B corresponds to 2, and f(2) = 3(2) + 1 = 7,

and so on.