## Inverse Functions

Table of Contents

## Definition

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

f^{-1}(y), wherey = f(x)

It says “finverse ofy“

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 isf^{-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 |

x^{2} <=> √y | x and y ≥ 0 |

n not zero (different rules when n is even, odd, positive, or negative) | |

e^{x} <=> ln(y) | y > 0 |

a^{x} <=> log(_{a}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 |