# Inverse Functions

Table of Contents

## Definition

The inverse is shown by putting a little “-1” after the function name, like this:

f-1(y), where y = f(x)

It says “f inverse of y

Here we have the function f(x) = 3x+2

The flow is

Inverse Functions make 3x + 2 back to x , substitute y = 3x + 2,

then

So, the inverse of f(x) = 3x + 2 is f-1(y) = Let’s put x = 3 on f(x):

f(3) = 3×3 + 2 = 11

Then put y = 11 on f-1(y):

f-1(11) = = 3

So,

f-1(y)=f-1(f(x)) = x

Then, how about f (f-1(x)) ?

Because f-1(y)= x, then f (f-1(y)) = f (x) = y

f (f-1(y)) = y. Which means f (f-1(x)) = x

So

f-1(f(x)) = f (f-1(x)) = x

## Inverses of Common Functions

So far is easy, because we know the inverse of Add is Subtract, and the inverse of Multiply is Divide. Ho about other functions?

### List of inverse of Multiply is Divide

 Inverse Note + <=> – x <=> ÷ Can’t divide by zero 1/x <=> 1/y x and y not zero x2 <=> √y x and y ≥ 0  n not zero (different rules when n is even, odd, positive, or negative) ex <=> ln(y) y > 0 ax <=> loga(y) y and a > 0 sin(x) <=> sin-1(y) -π/2 to +π/2 cos(x) <=> cos-1(y) 0 to π tan(x) <=> tan-1(y) -π/2 to +π/2

## Learn More

Set-Builder Notation

Intervals

Injective, Surjective & Bijective

Function Transformations

Sets Index