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Inverse Functions

NOTE f-1 (x) does NOT mean 1/f(x)

Inverse Functions

- Example Verify that f(x)-3x6 and g(x)-1/3x2

are inverses - Solution Find f(g(x)) and g(f(x)). If they

both equal x, then they are inverses.

g(f(x)) -1/3(-3x6)2 x-22 x

f(g(x)) -3(-1/3x2)6 x-66 x

ONE-TO-ONE FUNCTIONS

A function is one-to-one if for each y-value

there is only one x-value that can be paired with

it that is, no two ordered pairs have the same

second component.

Horizontal Line Test

- Used to determine whether a functions inverse

will be a function by seeing if the original

function passes the horizontal line test. - If the original function passes the horizontal

line test, then its inverse is a function. - If the original function does not pass the

horizontal line test, then its inverse is not a

function.

Finding the Inverse of a function

- The inverse of a function f can be found as

follows - Replace f (x) with y.
- Interchange x and y.
- Solve the resulting equation for y. If this

equation does not define y as a function of x,

the function does not have an inverse. Or

equivalently, the function is f is not a

one-to-one function and has no inverse - If f has an inverse, replace y in step 3 with f

-1 (x).

Sine Function Cosine Function

Inverse Trig Functions

- Goals
- Understand Domain/Range requirements for inverse

trig functions - Be able to calculate exact values for inverse

trig functions - Understand the composition of inverse trig

functions

The Inverse Sine Function

- The inverse sine function, also called the

arcsine function, is denoted by sin-1 or arcsin. - The function sin-1 has domain -1,1 and range

-p/2, p / 2 - In other words, y arcsin x, iff sin y x

The Inverse Cosine Function

- The inverse cosine function, also called the

arccosine function, is denoted by cos-1 or

arccos. - The function cos-1 has domain -1,1 and range

0, p - In other words, if y arccos x, iff cos y x

The Inverse Tangent Function

- The inverse tan function, also called the arctan

function, is denoted by tan-1 or arctan. - The function tan-1 has domain of all real numbers

and range -p/2 y p/2 - In other words, y arctan x, iff tan y x

tan (tan -1 x ) x for real numbers

tan -1(tan x ) x -p/2 x p/2

Summary

- With no restrictions, inverses of sin, cos and

tangent are not functions. - With restrictions
- Arcsin(x) ?
- Domain x-1 x 1
- Range? -p/2 ? p/2
- Arccos(x) ?
- Domain x-1 x 1
- Range ?0 ? p
- Arctan(x) ?
- Domain xx ? R
- Range ? -p/2 ? p/2

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