Question

a) Define the inverse of a function

b) When does a function have an inverse?

C) Does the function$\mathbf{f}\mathbf{\left(}\mathbf{n}\mathbf{\right)}\mathbf{=}\mathbf{10}\mathbf{-}\mathbf{n}$ from the set of integers to the set of integers have an inverse? If so, what is it?

Short answer

a) The inverse function of is$f^{-1}:B\to A$ such that$f^{-1}\left(x\right)=y$

b) When the function is a one-to-one correspondence.

c) Yes,$f^{-1}\left(n\right)=10-n$

Step-by-step solution

Step 1:Definitions

The functionis onto if and only if for every element $\mathbf{b}\mathbf{\in }\mathbf{B}$there exist an element $\mathbf{a}\mathbf{\in }\mathbf{A}$such that $\mathbf{f}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{=}\mathbf{b}$

The function f is one to one if and only if$\mathbf{f}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{b}\mathbf{\right)}$implies that a=b for all a and b in the domain.

is a one-to-one correspondence if and only ifis one to one and onto.

If f is a one-to-one correspondence, thenis invertible.

Inverse function

a)

Let$f:A\to B$ be a function.

Ifis a one-to-one correspondence, thenis invertible.

The inverse function of f is then$f^{-1}:B\to A$ such that${\mathrm{f}}^{-1}\left(\mathrm{x}\right)=\mathrm{y}$ if and only if$\mathrm{f}\left(\mathrm{y}\right)=\mathrm{x}$ (with $x\in A$and $x\in B$)

Invertible function

b)

A function has an inverse, when the function is a one-to-one correspondence.

Finding inverse of a function

c)

$f\left(n\right)=10-n$

The function f is one-to-one, because if $f\left(x\right)=f\left(y\right)$, then$10-x=10-y$ or equivalently x=y

The functionis onto, because if $x\in Z$, then $f(10-x)=10-f(10-x)=10-10+x=x$with$10-\mathrm{x}\in \mathrm{Z}$

Since,is one-to-one and onto f is a one-to-one correspondence and thus f has an inverse.

Since $\mathrm{f}\left(\mathrm{f}\right(\mathrm{x}\left)\right)=\mathrm{f}(10-\mathrm{x})=\mathrm{x}$,for all, $x\in Z$is the inverse of.