6.6: Inverse Functions

A bijection is a function that is both one-to-one and onto. Naturally, if a function is a bijection, we say that it is bijective. If a function \(f :A \to B\) is a bijection, we can define another function \(g\) that essentially reverses the assignment rule associated with \(f\). Then, applying the function \(g\) to any element \(y\) from the codomain \(B\), we are able to obtain an element \(x\) from the domain \(A\) such that \(f(x)=y\). Let us refine this idea into a more concrete definition.

Definition: inverse function

Let \(f :{A}\to{B}\) be a bijective function. Its inverse function is the function \({f^{-1}}:{B}\to{A}\) with the property that \[f^{-1}(b)=a \Leftrightarrow b=f(a).\] The notation \(f^{-1}\) is pronounced as “\(f\) inverse.” See Figure \(\PageIndex{1}\) for a pictorial view of an inverse function.

Figure \(\PageIndex{1}\): The pictorial view of an inverse function.

Why is \(f^{-1}:B \to A\) a well-defined function? For it to be well-defined, every element \(b\in B\) must have a unique image. This means given any element \(b\in B\), we must be able to find one and only one element \(a\in A\) such that \(f(a)=b\). Such an \(a\) exists, because \(f\) is onto, and there is only one such element \(a\) because \(f\) is one-to-one. Therefore, \(f^{-1}\) is a well-defined function.

If a function \(f\) is defined by a computational rule, then the input value \(x\) and the output value \(y\) are related by the equation \(y=f(x)\). In an inverse function, the role of the input and output are switched. Therefore, we can find the inverse function \(f^{-1}\) by following these steps:

1. Interchange the role of \(x\) and \(y\) in the equation \(y=f(x)\). That is, write \(x=f(y)\).
2. Solve for \(y\). That is, express \(y\) in terms of \(x\). The resulting expression is \(f^{-1}(x)\).

Be sure to write the final answer in the form \(f^{-1}(x) = \ldots\,\). Do not forget to include the domain and the codomain, and describe them properly.

Example \(\PageIndex{1}\label{invfcn-01}\)

To find the inverse function of \(f :{\mathbb{R}}\to{\mathbb{R}}\) defined by \(f(x)=2x+1\), we start with the equation \(y=2x+1\). Next, interchange \(x\) with \(y\) to obtain the new equation \[x = 2y+1. \nonumber\] Solving for \(y\), we find \(y=\frac{1}{2}\,(x-1)\). Therefore, the inverse function is \[{f^{-1}}:{\mathbb{R}}\to{\mathbb{R}}, \qquad f^{-1}(x)=\frac{1}{2}\,(x-1). \nonumber\] It is important to describe the domain and the codomain, because they may not be the same as the original function.

Example \(\PageIndex{2}\label{eg:invfcn-02}\)

The function \(s :{\big[-\frac{\pi}{2}, \frac{\pi}{2}\big]}\to{[-1,1]}\) defined by \(s(x)=\sin x\) is a bijection. Its inverse function is

\[s^{-1}:[-1,1] \to {\big[-\frac{\pi}{2}, \frac{\pi}{2}\big]}, \qquad s^{-1}(x)=\arcsin x. \nonumber\]

The function \(\arcsin x\) is also written as \(\sin^{-1}x\), which follows the same notation we use for inverse functions.

hands-on Exercise \(\PageIndex{1}\label{he:invfcn-01}\)

The function \(f :{[-3,\infty)}\to{[\,0,\infty)}\) is defined as \(f(x)=\sqrt{x+3}\). Show that it is a bijection, and find its inverse function

hands-on Exercise \(\PageIndex{2}\label{he:invfcn-02}\)

Find the inverse function of \(g :{\mathbb{R}}\to{(0,\infty)}\) defined by \(g(x) = e^x\).

Remark

Exercise caution with the notation. Assume the function \(f :{\mathbb{Z}}\to{\mathbb{Z}}\) is a bijection. The notation \(f^{-1}(3)\) means the image of 3 under the inverse function \(f^{-1}\). If \(f^{-1}(3)=5\), we know that \(f(5)=3\). The notation \(f^{-1}(\{3\})\) means the preimage of the set \(\{3\}\). In this case, we find \(f^{-1}(\{3\})=\{5\}\). The results are essentially the same if the function is bijective.

If a function \(g :{\mathbb{Z}}\to{\mathbb{Z}}\) is many-to-one, then it does not have an inverse function. This makes the notation \(g^{-1}(3)\) meaningless. Nonetheless, \(g^{-1}(\{3\})\) is well-defined, because it means the preimage of \(\{3\}\). If \(g^{-1}(\{3\})=\{1,2,5\}\), we know \(g(1)=g(2)=g(5)=3\).

In general, \(f^{-1}(D)\) means the preimage of the subset \(D\) under the function \(f\). Here, the function \(f\) can be any function. If \(f\) is a bijection, then \(f^{-1}(D)\) can also mean the image of the subset \(D\) under the inverse function \(f^{-1}\). There is no confusion here, because the results are the same.

Example \(\PageIndex{3}\label{eg:invfcn-03}\)

The function \(f :{\mathbb{R}}\to{\mathbb{R}}\) is defined as \[f(x) = \cases{ 3x & if $x\leq 1$, \cr 2x+1 & if $x > 1$. \cr} \nonumber\] Find its inverse function.

Solution

Since \(f\) is a piecewise-defined function, we expect its inverse function to be piecewise-defined as well. First, we need to find the two ranges of input values in \(f^{-1}\). The images for \(x\leq1\) are \(y\leq3\), and the images for \(x>1\) are \(y>3\). Hence, the codomain of \(f\), which becomes the domain of \(f^{-1}\), is split into two halves at 3. The inverse function should look like \[f^{-1}(x) = \cases{ \mbox{???} & if $x\leq 3$, \cr \mbox{???} & if $x > 3$. \cr} \nonumber\] Next, we determine the formulas in the two ranges. We find

\[f^{-1}(x) = \cases{ \textstyle\frac{1}{3}\,x & if $x\leq 3$, \cr \textstyle\frac{1}{2} (x-1) & if $x > 3$. \cr} \nonumber\] The details are left to you as an exercise.

hands-on Exercise \(\PageIndex{3}\label{he:invfcn-03}\)

Find the inverse function of \(g :{\mathbb{R}}\to{\mathbb{R}}\) defined by \[g(x) = \cases{ 3x+5 & if $x\leq 6$, \cr 5x-7 & if $x > 6$. \cr} \nonumber\] Be sure you describe \(g^{-1}\) properly.

Example \(\PageIndex{4}\label{eg:mod10fcn}\)

The function \(g :{\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}\) is defined by \(g(x)\equiv 7x+2\) (mod 10). Find its inverse function.

Solution

From \(x=g(y)\equiv7y+2\) (mod 10), we obtain \[y \equiv 7^{-1}(x-2) \equiv 3(x-2) \pmod{10}. \nonumber\] Hence, the inverse function \(g^{-1} :{\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}\) is defined by \(g^{-1}(x)\equiv 3(x-2)\) (mod 10).

hands-on Exercise \(\PageIndex{4}\label{he:invfcn-04}\)

The function \(h:{\mathbb{Z}_{57}}\to{\mathbb{Z}_{57}}\) defined by \(h(x)\equiv 49x-3\) (mod 57). Find its inverse function.

Example \(\PageIndex{5}\label{eg:invfcn-05}\)

Define \(h:{\mathbb{Z}_{10}}\to{\mathbb{Z}_{10}}\) according to \(h(x)=2(x+3)\bmod10\). Does \(h^{-1}\) exist?

Solution

Since \(2^{-1}\) does not exist, we suspect the answer is no. In fact, \(h(x)\) is always even, and it is easy to verify that \(\text{im}h = \{0,2,4,6,8\}\). Since \(h\) is not onto, \(h^{-1}\) does not exist.

Example \(\PageIndex{6}\label{eg:invfcn-06}\)

Find the inverse function of \(f :{\mathbb{Z}}\to{\mathbb{N}\cup\{0\}}\) defined by \[f(n) = \cases{ 2n & if $n\geq0$, \cr -2n-1 & if $n < 0$. \cr} \nonumber\]

Solution

In an inverse function, the domain and the codomain are switched, so we have to start with \(f^{-1}:\mathbb{N} \cup \{0\} \to \mathbb{Z}\) before we describe the formula that defines \(f^{-1}\). Writing \(n=f(m)\), we find \[n = \cases{ 2m & if $m\geq0$, \cr -2m-1 & if $m < 0$. \cr} \nonumber\] We need to consider two cases.

1. If \(n=2m\), then \(n\) is even, and \(m=\frac{n}{2}\).
2. If \(n=-2m-1\), then \(n\) is odd, and \(m=-\frac{n+1}{2}\).

Therefore, the inverse function is defined by \(f^{-1}:\mathbb{N} \cup \{0\} \to \mathbb{Z}\) by:

\[f^{-1}(n) = \cases{ \frac{2}{n} & if $n$ is even, \cr -\frac{n+1}{2} & if $n$ is odd. \cr} \nonumber\]

Verify this with some numeric examples.

hands-on Exercise \(\PageIndex{5}\label{he:invfcn-05}\)

The function \(f :{\mathbb{Z}}\to{\mathbb{N}}\) is defined as \[f(n) = \cases{ -2n & if $n < 0$, \cr 2n+1 & if $n\geq0$. \cr} \nonumber\] Find its inverse.

Let \(A\) and \(B\) be finite sets. If there exists a bijection \(f :{A}{B}\), then the elements of \(A\) and \(B\) are in one-to-one correspondence via \(f\). Hence, \(|A|=|B|\). This idea provides the basis for some interesting proofs.

Example \(\PageIndex{7}\label{eg:invfcn-07}\)

Let \(A=\{a_1,a_2,\ldots,a_n\}\) be an \(n\)-element sets. Recall that the power set \(\wp(A)\) contains all the subsets of \(A\), and \[\{0,1\}^n = \{(b_1,b_2,\ldots,b_n) \mid b_i\in\{0,1\} \mbox{ for each $i$, where $1\leq i\leq n$} \}. \nonumber\] Define \(F:{\wp(A)}\to{\{0,1\}^n}\) according to \(F(S) = (x_1,x_2,\ldots,x_n)\), where \[x_i = \cases{ 1 & if $a_i\in S$, \cr 0 & if $a_i\notin S$. \cr} \nonumber\] Simply put, \(F(S)\) is an ordered \(n\)-tuple whose \(i\)th entry is either 1 or 0, indicating whether \(S\) contains the \(i\)th element of \(A\) (1 for yes, and 0 for no).

It is clear that \(F\) is a bijection. For \(n=8\), we have, for example, \[F(\{a_2,a_5,a_8\}) = (0,1,0,0,1,0,0,1), \nonumber\] and \[F^{-1}\big((1,1,0,0,0,1,1,0)\big) = \{a_1,a_2,a_6,a_7\}. \nonumber\] The function \(F\) defines a one-to-one correspondence between the subsets of \(A\) and the ordered \(n\)-tuples in \(\{0,1\}^n\). Since there are two choices for each entry in these ordered \(n\)-tuples, we have \(2^n\) such ordered \(n\)-tuples. This proves that \(|\wp(A)|=2^n\), that is, \(A\) has \(2^n\) subsets.

hands-on Exercise \(\PageIndex{6}\label{he:invfcn-06}\)

Consider the function \(F\) defined in Example 6.6.7. Assume \(n=8\). Find \(F(\emptyset)\) and \(F^{-1}\big( (1,0,1,1,1,0,0,0)\big)\).

Summary and Review

• A bijection is a function that is both one-to-one and onto.
• The inverse of a bijection \(f :{A}{B}\) is the function \({f^{-1}}:{B}\to{A}\) with the property that \[f(x)=y \Leftrightarrow x=f^{-1}(y). \nonumber\]
• In brief, an inverse function reverses the assignment rule of \(f\). It starts with an element \(y\) in the codomain of \(f\), and recovers the element \(x\) in the domain of \(f\) such that \(f(x)=y\).

Exercise \(\PageIndex{1}\label{ex:invfcn-01}\)

Which of the following functions are bijections? Explain!

1. \(f :{\mathbb{R}}\to{\mathbb{R}}\), \(f(x)=x^3-2x^2+1\).
2. \(g :{[\,2,\infty)}\to{\mathbb{R}}\), \(g(x)=x^3-2x^2+1\).
3. \(h:{\mathbb{R}}\to{\mathbb{R}}\), \(h(x)=e^{1-2x}\).
4. \(p :{\mathbb{R}}\to{\mathbb{R}}\), \(p(x)=|1-3x|\).
5. \(q:{[\,2,\infty)}\to{[\,0,\infty)}\), \(q(x)=\sqrt{x-2}\).

Exercise \(\PageIndex{2}\label{ex:invfcn-02}\)

For those functions that are not bijections in the last problem, can we modify their codomains to change them into bijections?

Exercise \(\PageIndex{3}\label{ex:invfcn-03}\)

Let \(f\) and \(g\) be the functions from \((1,3)\) to \((4,7)\) defined by \[f(x) = \frac{3}{2}\,x+\frac{5}{2}, \qquad\mbox{and}\qquad g(x) = -\frac{3}{2}\,x+\frac{17}{2}. \nonumber\] Find their inverse functions. Be sure to describe their domains and codomains.

Exercise \(\PageIndex{4}\label{ex:invfcn-04}\)

Find the inverse function \(f :{\mathbb{R}}\to{\mathbb{R}}\) defined by \[f(x) = \cases{ 3x+5 & if $x\leq 6$, \cr 5x-7 & if $x > 6$. \cr} \nonumber\]

Be sure you describe \(f^{-1}\) correctly and properly.

Exercise \(\PageIndex{5}\label{ex:invfcn-05}\)

The function \(g :{[\,1,3\,]}\to{[\,4,\,7]}\) is defined according to \[g(x) = \cases{ x+3 & if $1\leq x< 2$, \cr 11-2x & if $2\leq x\leq 3$. \cr} \nonumber\] Find its inverse function. Be sure you describe it correctly and properly.

Exercise \(\PageIndex{6}\label{ex:invfcn-06}\)

Find the inverse of the function \(r :{(0,\infty)}\to{\mathbb{R}}\) defined by \(r(x)=4+3\ln x\).

Exercise \(\PageIndex{7}\label{ex:invfcn-07}\)

Find the inverse of the function \(s :{\mathbb{R}}\to{(-\infty,-3)}\) defined by \(s(x)=4-7e^{2x}\).

Exercise \(\PageIndex{8}\label{ex:invfcn-08}\)

Find the inverse of each of the following bijections.

1. \(h:{\{1,2,3,4,5\}}\to{\{a,b,c,d,e\}}\), \(h(1)=e\), \(h(2)=c\), \(h(3)=b\), \(h(4)=a\), \(h(5)=d\).
2. \(k :{\{1,2,3,4,5\}}\to{\{1,2,3,4,5\}}\), \(k(1)=3\), \(k(2)=1\), \(k(3)=5\), \(k(4)=4\), \(k(5)=2\).

Exercise \(\PageIndex{9}\label{ex:invfcn-09}\)

Find the inverse of each of the following bijections.

1. \(u:{\mathbb{Q}}\to{\mathbb{Q}}\), \(u(x)=3x-2\).
2. \(v:{\mathbb{Q}-\{1\}}\to{\mathbb{Q}-\{2\}}\), \(v(x)=\frac{2x}{x-1}\).
3. \(w:{\mathbb{Z}}\to{\mathbb{Z}}\), \(w(n)=n+3\).

Exercise \(\PageIndex{10}\label{ex:invfcn-10}\)

Find the inverse of each of the following bijections.

1. \(r :{\mathbb{Z}_{12}}\to{\mathbb{Z}_{12}}\), \(r(n)\equiv 7n\) (mod 12).
2. \(s :{\mathbb{Z}_{33}}\to{\mathbb{Z}_{33}}\), \(s(n)\equiv 7n+5\) (mod 33).
3. \(t :{\mathbb{Z}}\to{\mathbb{N}\cup\{0\}}\), \(t(n) = \cases{ 2n-1 & if $n > 0$, \cr -2n & if $n\leq0$,\cr}\)

Exercise \(\PageIndex{11}\label{ex:invfcn-11}\)

The images of the bijection \({\alpha}:{\{1,2,3,4,5,6,7,8\}}\to{\{a,b,c,d,e,f,g,h\}}\) are given below. \[\begin{array}{|c||*{8}{c|}} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \alpha(x)& g & a & d & h & b & e & f & c \\ \hline \end{array} \nonumber\] Find its inverse function.

Exercise \(\PageIndex{12}\label{ex:invfcn-12}\)

Below is the incidence matrix for the bijection \({\beta}: {\{a,b,c,d,e,f\}}\to{\{x,y,z,u,v,w\}}\). \[\begin{array}[t]{cc} & \begin{array}{cccccc} u & v & w & x & y & z \end{array} \\ \begin{array}{c} a \\ b \\ c \\ d \\ e \\ f \end{array} & \left(\begin{array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right) \end{array} \nonumber\] Find its inverse function.