Inverse Functions - Notesformsc

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In the previous article, you learned about composite function, in this article, you will learn about inverse functions. The term “inverse” means to “undo” something and which is what the “inverse” of a function do. If a function $f(x)$ find the $y$ value for an $x$ value, the inverse of the function does the opposite, meaning it finds the $x$ value for a given $y$ value.

Function and its inverse

Remember from previous articles, that the function $f(x)$ is set of ordered pair $(x. y)$ , that is, for every input $x$ there is a one and only $y$ .

Figure 1 - Function f(x) is an ordered pair such as (x1, y1), (x2, y2) — Figure 1 – Function f(x) is an ordered pair such as (x1, y1), (x2, y2)

Suppose $g(x)$ is function that takes input $y$ and gives us the value $x$ . We are just undoing the function $f(x)$ . Therefore, if $g(x)$ is inverse function of $f(x)$ , then it is denoted as $f^{-1}(x)$ .

Figure 2 - The function g(x) is inverse function with ordered pair (y1, x1), (y2, x2). — Figure 2 – The function g(x) is inverse function with ordered pair (y1, x1), (y2, x2).

The inverse of the function $f^{-1}(x)$ is the set of ordered pairs $(y, x)$ .

One-To-One Relationship

If you look at the figure 2, you will find that there is a one-to-one relationship between function $f(x)$ and its inverse $g(y) = f^{-1}(x)$ .

If there are two functions $f(x)$ and its inverse function $f^{-1}(x)$ , then

$\begin{align*}f(f^{-1}(x)) = x\end{align}$

Where $x$ is all values in the domain of inverse function $f^{-1}(x)$ . Similarly,

$\begin{align*}f^{-1}(f(x)) = x\end{align}$

Where $x$ is all values in the domain of function $f(x)$ .

If a function $f(x)$ does not maintain the one-to-one relationship, there is no inverse. Consider the quadratic function $f(x) = x^2$ . The function has same outputs for $f(-2) = 4$ and $f(2) = 4$ .

Figure 3 - Inverse is not a function — Figure 3 – Inverse is not a function

If we take an inverse of the function which is $f^{-1}(4) = \pm2$ implies that there are two values for $f^{-1}(4)$ . Therefore, inverse is not a function because a function can only have one output for given input $x$ .

Horizontal Line Test

The easiest way to understand whether a function has inverse or not is to perform a horizontal line test on the graph of the function. To understand this concept , we will use our previous example of quadratic function $f(x) = x^2$ . The graph of function is given below.

We perform a horizontal line test , that is, draw a horizontal line and if the line intersect the graph of function $f(x) = x^2$ it has no inverse function. The figure 4 above, shows that the horizontal line intersect the graph of parabola, at $(-2, 4)$ and $(2, 4)$ . Therefore, $f(x) = x^2$ has no inverse.

Graph of Inverse Function

The graph of inverse function, if exists, can be obtained easily by changing the set of ordered pair $(x, y)$ to the set of ordered pair $(y, x)$ .

For example, consider the graph of exponential function $f(x)=2^x$ which is an exponential function. The function passes the horizontal line test, therefore, an inverse function exists.

x	f(x) =2^x
-4	0.0625
-3	0.125
-2	0.25
-1	0.5
0	1
1	2
2	4
3	8

Values for the exponential function f(x) = 2^x

Figure 5 – Graph of exponential function

The inverse function is set of all ordered pairs $(y, x)$ . The inverse function of exponential function is $f^{-1}(x) = log_2(x)$ .

Figure 6 – Inverse of exponential function is the logarithmic function f*{-1}(x) = log_2(x)

x	f^{-1}(x)
0.0625	-4
0.125	-3
0.25	-2
0.5	-1
1	0
2	1
4	2
8	3

Values for logarithmic functions

Note that the inverse function is accepting all positive $x$ values, and all inverse function (shown in red) will reflect over the line ( in green) $y = x$ . The ordered pairs $(x, y)$ in exponential function is replaced with ordered pair $(y , x)$ .

How To Find The Inverse Function ?

To find the inverse of the function you must follow the following steps. If $f(x)$ is a function with an expression, then

Write $y$ instead of $f(x)$ .
Interchange the $x$ and $y$ in the equation.
Solve the equation for $y$ and if the equation does not define $y$ in terms of $x$ , then there is no inverse. Otherwise, you will have an equation that defines $y$ in terms of $x$ .
Replace $y$ with $f^{-1}(x)$ .

Now, it is necessary to verify the inverse function, that can be done by verifying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ . The composition of $(f \circ f^{-1})$ and $(f^{-1} \circ f(x))$ is an algebraic proof of inverse function.

Example #1

Find the inverse function of $f(x) =2^{x+1}$ .

Solution:

The given equation is $f(x) =2^{x+1}$ . exponential equation and we expect to find a logarithmic function as inverse. But, we will go through all the steps to find the inverse of this function.

$\begin{align*} &f(x) =2^{x+1}\\ \\ &Step1\\ & y = 2^{x + 1}\\ \\ &Step2\\ &x = 2^{y + 1} \\ \\ &Step3\\ & x = 2^y \cdot 2\\ \\ & x/2 = 2^y\\ \\ & 2^y = x /2\\ \\ & Take\hspace{2mm} log \hspace{2mm} with \hspace{2mm} base \hspace{2mm} 2 \hspace{2mm} of \hspace{2mm} both \hspace{2mm} \\ \\ & \log_2 (2^y) = \log_2(x / 2)\\ \\ & y = \log_2(x) - \log_2(2) \hspace{1 cm} By\hspace{2mm} Rule \hspace{2mm}\log_a(M/N) = \log_a(M) - log_a(N)\\ \\ & y = \log_2(x) - 1 \hspace{1 cm} By\hspace{2mm} Rule \hspace{2mm} \log_a(a) = 1\\ \\ &Step4\\ & f^{-1}(x) = \log_2(x) - 1\\ \end{align}$

Now , we must verify if the inverse function is correct, by using composition of functions. Therefore,

$\begin{align*}&f^{-1}(f(x)) = \log_2(2^{x + 1} ) - 1\\ \\ &x =\log_2(2^{x + 1} ) - 1 \hspace{1 cm} By \hspace{2mm} Rule \hspace{2mm} log_a(a^x) = x\\ \\ &x =( x +1 ) - 1\\ \\ &x = x + 1 - 1\\ \\ & x = x \end{align}$

Similarly,

$\begin{align*}&f(f^{-1}(x)) = \log_2(2^{x + 1}) - 1\\ \\ &x = \log_2(2^{x + 1}) - 1\\ \\ &x = (x + 1) - 1 \hspace{1 cm} By \hspace{2mm} Rule \hspace{2mm}log_a(a^x) = x \\ \\ &x = x + 1 - 1\\ \\ &x = x\\ \\ \end{align}$

Example #2

Find the inverse function of the function $f(x) = 7x - 3$ .

Solution:

The function $f(x) = 7x - 3$ is a linear function. Therefore, an inverse function exists, but we will verify whether inverse exist of not by following our steps to find the inverse of a function.

$\begin{align*}&f(x) = 7x - 3\\ \\ &Step \hspace{2mm}1 \\ &y = 7x - 3\\ \\ &Step \hspace{2mm} 2\\ &x = 7y - 3\\ \\ &Step \hspace{2mm} 3\\ &x + 3 = 7y\\ \\ &7y = x + 3\\ \\ &y = \frac{x + 3}{7}\\ \\ Step \hspace{2mm}4\\ &f^{-1} = \frac{x + 3}{7} \end{align}$

Now, we must verify the inverse function.

$\begin{align*} &f^{-1}(f(x)) = \left (\frac{7x - 3 + 3}{7}\right ) \\ \\ &x = \frac{7x}{7}\\ \\ &x = x \end{align}$

Similarly,

$\begin{align*} &f(f^{-1}(x)) = 7\left (\frac{x + 3}{7}\right ) - 3\\ \\ &x = (x + 3) - 3\\ \\ &x = x + 3 - 3\\ \\ &x = x \end{align}$

Restricted Domain

It is possible to find an inverse function to functions that does not have any inverse if we restrict the domain. It means we only accept set of ordered pairs $(x, y)$ of function $f(x)$ for which there exist ordered pairs $(y, x)$ .

Consider the graph of absolute value function [latedpage] $f(x) = |x|$ . This graph did not pass the horizontal line test and for each $y$ value, there exist two $x$ values.

Figure 7 - Graph of absolute function is f(x) = |x| and it fails the horizontal line test. — Figure 7 – Graph of absolute function is f(x) = |x| and it fails the horizontal line test.

The graph will pass the horizontal line test if we restrict the domain to $x \geq 0$ , therefore, an inverse of absolute value function exists if we choose values between the interval $[0, \infty]$ .

Figure 8 - Graph of absolute function with restricted domain. — Figure 8 – Graph of absolute function with restricted domain.

The inverse of the new restricted absolute function is as follows.

$\begin{align*} &y = |x|\\ \\ &x = |y|\\ \\ &|x| = || y || \\ & y = |x| where \hspace{2mm} x \geq 0\\\\ & f^{-1}(x) = |x| \end{align}$

We must observe two things,

The absolute value cannot be a negative number, therefore, $f(x) = |x|$ where $x \geq 0$ .
The function $f(x)$ and its inverse does not reflect over $x = y$ , but $f(x) = f^{-1}(x) = |x| = x = y$ .