Inverse functions

The function $g$ is the inverse of the one-to-one function $f$ if and only if the following are true:

$g(f(x))=x\,$

$f(g(x))=x\,$

The inverse of function $f$ is denoted as $f^{-1}$ .

Geometrically $f^{-1}$ is the reflection of $f$ across the line $y=x$. Conceptually, using the box analogy, a function's inverse box undoes what the function's regular box does.

Example:

$f(x)=2x\,$
$f^{-1}(x)={\frac {1}{2}}x\,$

$f(f^{-1}(x))=f({\frac {1}{2}}x)\,$
$f(f^{-1}(x))=2({\frac {1}{2}}x)\,$
$f(f^{-1}(x))=x\,$


$f^{-1}(f(x))=f^{-1}(2x)\,$
$f^{-1}(f(x))={\frac {1}{2}}(2x)\,$
$f^{-1}(f(x))=x\,$

To find the inverse of a function, remember that when we use $f^{-1}(x)$ as an input to $f$ the result is $x$. So start by writing $x=f\left(f^{-1}(x)\right)$ and solve for $f^{-1}$

Example:

Suppose:$f(x)=2x+1\,$
Then $x=f\left(f^{-1}(x)\right)$
$x=2f^{-1}(x)+1\,$
$x-1=2f^{-1}(x)\,$
$f^{-1}(x)={\frac {x-1}{2}}\,$

The Domain of an inverse function is exactly the same as the Range of the original function. If the Range of the original function is limited in some way, the inverse of a function will require a restricted domain.

Example:

$f(x)={\sqrt {x-1}}\,$

$x=f\left(f^{-1}(x)\right)$

$x={\sqrt {f^{-1}(x)-1}}\,$

$x^{2}={\sqrt {f^{-1}(x)-1}}^{2}\,$

$x^{2}=f^{-1}(x)-1$

$f^{-1}(x)=x^{2}+1\,$

The Range of $f(x)$ is $f(x)\geq 0$. So the Domain of $f^{-1}(x)$ is $x\geq 0$.

One-to-one function

A function that for every input there exists an output unique to that input.

Equivalently, we may say that a function $f$ is called one-to-one if for all $x,x'\in A,f(x)=f(x')$ implies that $x=x'$ where A is the domain set of f and both x and x' are members of that set.

Horizontal Line Test
If no horizontal line intersects the graph of a function in more than one place then the function is a one-to-one function.

The Existence of Inverse Functions

A function has an inverse if and only if it is one-to-one: that is, if for each y-value there is only one corresponding x-value. To test whether or not a function is one-to-one, we can draw multiple horizontal lines through the graph of the function. If any of these horizontal lines intersects with the graph of the function more than once, then the function is NOT one-to-one.

For example, take the graph of the function $f:\mathbb {R} \to \mathbb {R} ,f(x)=x^{2}$.

We can draw a horizontal line anywhere on the graph (except for the turning point at $x=0$) which will intersect with the graph twice. Therefore f is NOT a one-to-one function, and will NOT have an inverse function.

However, consider the function $g:\mathbb {R} \to \mathbb {R} ,g(x)=x^{3}$.

This function IS one-to-one, and will therefore have an inverse function, which we label $g^{-1}\,$.

Resources

• Inverse Functions. Written notes created by Professor Esparza, UTSA.
• Functions, Wikibooks: Algebra
• Inverse Functions, Wikibooks: VCE Mathematical Methods

Licensing

Content obtained and/or adapted from:

• Functions, Wikibooks: Algebra under a CC BY-SA license
• Inverse Functions, Wikibooks: VCE Mathematical Methods under a CC BY-SA license