Algebra/Chapter 11/Inverses of Functions

Inverse function

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 $x\geq 0$.

When functions were introduced in this chapter they were described as a special kind of relation. A relation is simply a connection between one number and some others. On the other hand a function has just one connection between a number and another number. For instance absolute value is a function because the absolute value of any number will map onto one, and only one number. On the other hand the square root operation is a relationship. ${\sqrt {x}}^{2}$ can be true for $x$ or $-x$.

We defined the identity value for addition as 0 and for multiplication as 1. This is because adding zero to a number does not change that number, or multiplying a number by 1 does not change that number. We defined the inverse of addition as subtraction. This is because a - a =0. The inverse of multiplication is division (except for 0) since a / a = 1. We define the inverse of a function as the function g such that g(f(x)) = x.

For example the inverse of the squared function is the square root function for whole numbers. That is for zero and non-negative rational numbers the ${\sqrt {x}}^{2}=x$. This is not true for the real numbers. For real numbers ${\sqrt {x}}^{2}=x\ or-x$.

When explaining the definition of domain and range of a relation we defined the vertical line test as stating that on a Cartesian plane if we let the x axis indicate the domain of a function and the y axis indicate the domain than the relationship is a function if the graph is not intersected more than once by a vertical line at every point on the graph. The vertical line test is an informal way to see if a relationship is a function. We can also use it to determine if the function has an inverse. For instance if we let $y=x^{2}$ we can see from its graph that it is a function. On the other hand if we graph $y={\sqrt {x}}$ we can see that a vertical line will cross the graph twice for every value except 0.