Informally, two functions $f$ and $g$ are inverses if each reverses, or undoes, the other. More precisely:

Definition 9.1.1 Two functions $f$ and $g$ are inverses if for all $x$ in the domain of $g$, $f(g(x))=x$, and for all $x$ in the domain of $f$, $g(f(x))=x$.

Example 9.1.2 $f=x^3$ and $g=x^{1/3}$ are inverses, since $\ds (x^3)^{1/3}=x$ and $(x^{1/3})^3=x$.

Example 9.1.3 $f=x^2$ and $g=x^{1/2}$ are not inverses. While $(x^{1/2})^2=x$, it is not true that $(x^2)^{1/2}=x$. For example, with $x=-2$, $((-2)^2)^{1/2}=4^{1/2}=2$.

The problem in the previous example can be traced to the fact that there are two different numbers with square equal to 4. This turns out to be precisely descriptive of functions without inverses.

Definition 9.1.4 Let $A$ and $B$ be sets and let $f:A\to B$ be a function. We say that $f$ is injective or one-to-one if $f(x)=f(y)$ implies that $x=y$.

We say that $f$ is surjective or onto if for every $b\in B$ there is an $a\in A$ such that $f(a)=b$.

If $f$ is both injective and surjective, then $f$ is bijective or one-to-one and onto.

We are interested only in the case that $A$ and $B$ are sets of real numbers, and in this case there is a nice geometric interpretation of injectivity. It is often easy to use this interpretation to decide whether a function is or is not 1-1.

Theorem 9.1.5 (Horizontal line test) If $f$ is a function defined on some subset of the real numbers, then $f$ is injective if and only if every horizontal line intersects the graph of $f$ at most once.

Example 9.1.6 The function $f=x^2$ fails this test: horizontal lines $y=k$ for $k>0$ intersect the graph of $f$ twice. (The horizontal line $y=0$ does intersect it only once, and lines $y=k$, $k< 0$, do not intersect the graph at all.)

Example 9.1.7 In each of these cases, we assume that $f\colon A\to \R$, where $A$ is the set of all real numbers for which $f$ makes sense.

The function $f(x)=x$ is bijective.

The function $f(x) = x^2$ is neither injective nor surjective. If we think of $f$ as a function from $\R$ to the non-negative real numbers, then $f$ is surjective; in other words, if a function is not surjective this is not a major stumbling block.

The function $f(x) =1/x$ is injective but not surjective since there is no value of $x$ such that $f(x)=0$.

The function $f(x) = x(x-1)(x+1)$ is surjective but not injective; $f(x) =0$ for three different values of $x$. On the other hand $\ds\lim_{x\to \infty} f(x)=\infty$ and $\ds\lim _{x\to -\infty} f(x)=-\infty$. Since $f$ is continuous on $\R$, the intermediate value theorem (2.5.6) guarantees that $f$ takes all values between $-\infty$ and $\infty$.

The derivative furnishes us with a convenient criterion for injectivity without explicitly looking for points where injectivity may fail.

Theorem 9.1.8 If $f$ is differentiable and $f'(x) >0$ then $f$ is injective.

Proof.
Suppose that $f(a) =f(b)$ for some $a< b$. By Rolle's theorem (6.5.1) there exists $c\in (a,b)$ such that $f'(c) =(f(b) -f(a))/(b-a)=0$, which contradicts the hypothesis that $f'(x) >0$. Hence, if $f(a)=f(b)$ then $a=b$.

$\square$

In the same way, we can see that if $f'(x)< 0$ then $f$ is injective.

Example 9.1.9 Let $f(x) =x^5 + x$. Since $f'(x) = 5x^4 + 1 >0$, $f$ is injective.

Example 9.1.10 Let $f(x) =2x+\sin x$. Then $f'(x) = 2+\cos x \geq 1$ for every $x$. Hence, $f$ is injective.

Example 9.1.11 Let $f(x) =x^3$. This $f$ is injective although the above theorem does not apply, since $f'(0)=0$. Therefore, the conditions in the theorem are sufficient but not necessary.

Our knowledge of derivatives can also lead us to conclude that a function is not injective.

Theorem 9.1.12 If $f$ is continuous and has a local maximum or minimum then $f$ is not injective.

Proof.
Suppose that $f$ has a local maximum at $x=c$. Then in some interval $(c-h, c+h)$, $f(x)\le f(c)$. Let $a\in (c-h, c)$. If $f(a) =f(c)$ then $f$ is not injective; otherwise, $f(a) < f(c)$.

Let $b\in (c,c+h)$. If $f(b) = f(c)$ or $f(b) =f(a)$ then $f$ is not injective. Otherwise, either $f(b) < f(a)< f(c)$ or $f(a) < f(b)< f(c)$. If $f(b) < f(a)$ then by the intermediate value theorem, there is a number $d\in (c, b)$ such that $f(d) =f(a)$ and so $f$ is not injective. Likewise, if $f(a) < f(b)$ then there is a number $d$ in $(a,c)$ such that $f(d) = f(b)$ and so $f$ is not injective.

In every case, we see that $f$ is not injective.

$\square$

To return to our principal interest, inverse functions, we now connect bijections and inverses.

Theorem 9.1.13 Suppose $f\colon A\to B$ is a bijection. Then $f$ has an inverse function $g\colon B\to A$.

Proof.
Suppose $b\in B$. Since $f$ is onto, there is an $a\in A$ such that $f(a)=b$. Since $f$ is 1–1, $a$ is the only element of $A$ with this property. We let $g(b)=a$. Now it is easy to see that for all $a\in A$, $g(f(a))=a$ and for all $b\in B$, $f(g(b))=b$.

$\square$

We really don't have any choice about how to define $g$ in this proof; if $f$ is a bijection, its inverse is completely determined. Thus, instead of using a new symbol $g$, we normally refer to the inverse of $f$ as $f^{-1}$.

Unfortunately, it is often difficult to find an explicit formula for the inverse of a given function, $f$, even if it is known that $f$ is bijective. Generally, we attempt to find an inverse in this way:

1. Write $y=f(x)$.

2. Interchange $x$ and $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Step 3 is the hard part; indeed it is sometimes impossible to perform using algebraic operations.

Example 9.1.14 Find the inverse of $f(x) =(2x-6)/(3x+7)$. First we write $x=(2y-6)/(3y+7)$. Now we solve for $y$: \eqalign{ x&= {2y-6\over3y+7}\cr x(3y+7) &= 2y-6\cr 3xy+7x &= 2y - 6\cr 7x+6 &= 2y-3xy\cr 7x+6 &= y(2-3x)\cr {7x+6\over 2-3x} &= y\cr } Finally, we say $f^{-1} (x) =(7x+6)/(2-3x)$

Example 9.1.15 Find the inverse function of $f(x) =x^2 - 4x + 8$ where $x \geq 2$. What are the domain and range of the inverse function?

First, $y =x^2 - 4x + 8$ becomes $x=y^2-4y+8$. Now we complete the square: $x=(y-2)^2 +4$ and rearrange to get $x-4=(y-2)^2$. Since in the original function $x-2\ge 0$, and we have switched $x$ and $y$, we know that $y-2\ge 0$. Thus taking the square root, we know $y-2=\sqrt{x-4}$, {\bf not} $y-2=-\sqrt{x-4}$. Finally we write $y=f^{-1}(x)=2+\sqrt{x-4}$. The domain of $f$ is $x\geq 4$ and the range is $y\geq 2$.

While simple in principle, this method is sometimes difficult or impossible to apply. For example, consider $f(x)=x^3 + x$. Since $f'(x) = 3x^2 +1 >0$ for every $x$, $f$ is injective. (In fact it is bijective.) To find the inverse as above, we would need to solve $x=y^3+y$ for $y$; while possible, this is considerably more difficult than solving the quadratic of the previous example. Some simple looking equations are impossible to solve using algebraic manipulation.

For example, consider $f(x) =x^5 + x^3 + x +1$ a "quintic'' polynomial (i.e., a fifth degree polynomial). Since $f'(x)= 5x^4 + 3x^2 + 1>0$, $f$ is injective (and indeed $f$ is bijective). If there were a quintic formula, analogous to the quadratic formula, we could use that to compute $f^{-1}$. Unfortunately, no such formula exists—fifth degree equations cannot in general be solved. (There are exceptions; $x^5=1$ can be solved, for example.)

Fortunately, it is often more important to know that a function has an inverse then to be able to come up with an explicit formula. Once an inverse is known to exist, numerical techniques can often be employed to obtain approximations of the inverse function. Thus, theorem 9.1.8 and proposition 9.1.12 provide useful criteria for deciding whether a function is invertible.

We now turn to the calculus of inverse functions.

Theorem 9.1.16 Let $A$ be an open interval and let $f:A\to \R$ be injective and continuous. Then $f^{-1}$ is continuous on $f(A)$.

Proof.
Since $A$ is an open interval and $f$ is injective and continuous it follows by proposition 9.1.12 that $f$ has no local maxima or minima. Hence, $f$ is either strictly increasing or strictly decreasing. Without loss of generality, $f$ is strictly increasing.

Fix $b\in f(A)$ . Then there exists a unique $a\in A$ such that $f(a) = b$. Let $\epsilon > 0$ and we may assume that $(a-\epsilon , a+\epsilon ) \subseteq A$. Let $\delta =\min \{ b- f(a-\epsilon ) , f(a+\epsilon ) -b \}$ and note that $\delta >0$ since $f$ is increasing. Then the interval $(b-\delta , b+\delta )$ is mapped by $f^{-1}$ into $(a-\epsilon , a+\epsilon )$. Since $\epsilon$ was arbitrary, it follows that $f^{-1 }$ is continuous at $b$.

$\square$

Our principal interest in inverses is the simple relationship between the derivative of a function and its inverse.

Theorem 9.1.17 (Inverse function theorem) Let $A$ be an open interval and let $f:A\to \R$ be injective and differentiable. If $f'(x) \neq 0$ for every $x\in A$ then $f^{-1}$ is differentiable on $f(A)$ and $(f^{-1})'(x) = 1/f'(f^{-1}(x))$.

Proof.
Fix $b\in f(A)$. Then there exists a unique $a\in A$ such that $f(a)=b$. For $y\neq b$, let $x=f^{-1} (y)$. Since $f$ is differentiable, it follows that $f$ and hence $f^{-1}$ are continuous.

Then $$\lim_{y\to b}{f^{-1} (b) - f^{-1} (y)\over b-y} = \lim_{x\to a}{a-x\over f(a) -f(x)} ={1\over f'(a)}.$$

$\square$

In Leibniz notation, this can be written as $\ds{dx\over dy} ={1\over dy/dx}$, which is easy to remember since it looks like ordinary fractional algebra.

Example 9.1.18 Let $f(x) = 3x^3 + 5x - 7$. Since $f(0)=-7$, $f^{-1} (-7) =0$. Since $f'(x) = 9x^2 + 5$, $f'(0)=5$ and so $(f^{-1})' (-7)=1/f'(0)=1/5$.

Exercises 9.1

Ex 9.1.1 Which of the following functions are injective? Which are surjective? Which are bijective? Sketch the graph of each function to illustrate your answers.

a. $f\colon \R\to\R$, $f(x)=x^2$.

b. $f\colon [0, \infty )\to\R$, $f(x)=x^2$.

c. $f\colon \R\to[0, \infty )$, $f(x)=x^2$.

d. $f\colon (-\infty,0]\to[0, \infty )$, $f(x)=x^2$.

Ex 9.1.2 Which of the following functions are injective? Which are surjective? Which are bijective? Sketch the graph of each function to illustrate your answers.

a. $f\colon \R\to\R$, $f(x)=x^3$.

b. $f\colon [0, \infty )\to\R$, $f(x)=\sqrt{x}$.

c. $f\colon \R\to[-1, 1]$, $f(x)=\sin x$.

d. $f\colon [0,\pi]\to[-1, 1]$, $f(x)=\cos x$.

Ex 9.1.3 Define $$f(x)=\cases{ \ds-{1\over x}&x\neq 0\cr \ds 1&x=0\cr}$$

Show that $f$ is not injective on $\R$. Show that $f'(x) >0$ for $x\neq 0$. Why does this not contradict theorem 9.1.8?

Ex 9.1.4 Define $$f(x)=\cases{ \ds 1-x&-2\leq x< 0\cr \ds 0&x=0\cr \ds 10+x&0< x \leq 2\cr}$$

Show that $f$ is injective and has a local minimum. Why does this not contradict theorem 9.1.12?

Ex 9.1.5 If $A=\R$ sketch the graph of the identity function on $A$.

Ex 9.1.6 Find the inverse function of $f(x) =(4x-6)/(7x+ 5)$. What are the domain and range of $f^{-1}$?

Ex 9.1.7 Find the inverse function of $f(x) = 11x/(13x-6)$. What are the domain and range of $f^{-1}$?

Ex 9.1.8 Find the inverse function of $f(x)=ax+b$ when $a\neq 0$. What are the domain and range of $f^{-1}$?

Ex 9.1.9 Find the inverse function of $f(x) = 1/(cx+d)$ when $c\neq 0$. What are the domain and range of $f^{-1}$?

Ex 9.1.10 Suppose that $ad-bc \neq 0$. Find the inverse function of $f(x) = (ax+b)/(cx+d)$. What are the domain and range of $f^{-1}$? (The domain and range will depend on which if any of $a,b,c$, and $d$ are zero.)

Note: The condition $ad-bc \neq 0$ is a technical condition which ensures that both the domain of $f$ will be all real numbers with perhaps one exception and that that the range of $f$ will be all real numbers with perhaps one exception.

Ex 9.1.11 Find the inverse function of $f(x) =|x-4|$ for $x\leq4$. What are the domain and range of $f^{-1}$?

Ex 9.1.12 Find the inverse function of $f(x) = \sqrt{x-5}$. What are the domain and range of $f^{-1}$?

Ex 9.1.13 Find the inverse function of $f(x) = x^3 - 5$. What are the domain and range of $f^{-1}$?

Ex 9.1.14 Find the inverse function of $f(x) =x^7 - 2$. What are the domain and range of $f^{-1}$?

Ex 9.1.15 Find the inverse function of $f(x) =2x^2 +8x - 4$ for $x\geq -2$. What are the domain and range of $f^{-1 }$?

Ex 9.1.16 Find the inverse function of $f(x) =x^2 -9x + 10$ for $x\leq 3$. What are the domain and range of $f^{-1 }$?

Ex 9.1.17 Find the inverse function of $f(x)= x^2 +bx+ c$ for $x\geq -b/2$. What are the domain and range of $f^{-1 }$?

Ex 9.1.18 Find the inverse function of $f(x)= x^2 +bx+ c$ for $x\leq -b/2$. What are the domain and range of $f^{-1}$?

Ex 9.1.19 Find the inverse function of $f(x) = (1+\sqrt{x})/(1-\sqrt{x})$. What are the domain and range of $f^{-1 }$?

Ex 9.1.20 Show that $f(x) =x^7 + 3x$ has an inverse function on $\R$.

Ex 9.1.21 Show that $f(x) =x^{19/9} +x^5$ has an inverse function on $\R$.

Ex 9.1.22 Let $A\subset \R$ and let $f: A\to\R$ be injective. Then $f^{-1}$ exists on $f(A)$.

Note that the point $P(a, f(a))$ is on the graph of $f$ and that $Q(f(a), a)$ is the corresponding point on the graph of $f^{-1}$.

a. Show that if $a\neq f(a)$ then the slope of the line segment $PQ$ is $-1$.

b. Conclude that if $a\neq f(a)$ the line segment $PQ$ is perpendicular to the graph, $L$, of the identity function on $\R$.

c. Show that the midpoint of $PQ$ is on $L$.

d. Conclude that the graph of $f^{-1}$ is the graph of $f$ reflected through $L$.

Ex 9.1.23 Let $f(x) =x^3 +x$. Sketch the graph of $f$ and $f^{-1}$ on the same diagram.

Ex 9.1.24 Let $f(x) =x^5 + x^3 +1$. Sketch the graph of $f$ and $f^{-1}$ on the same diagram.

Ex 9.1.25

a. Suppose that $f$ is an increasing function on $\R$. What can you say about $f^{-1}$?

b. Suppose that $f$ is a concave up function on $\R$. What can you say about $f^{-1}$?

In both parts, use exercise 22 to illustrate your claim.

Ex 9.1.26 Let $f(x) = 3x^3 + 9x + 4$. Compute $(f^{-1})'(4)$.

Ex 9.1.27 Let $f(x) = 2x^2 + 11$. Show that $f$ is increasing at $x=0$. Thus, there is an interval $I$ containing $0$ such that $f$ is injective on $I$. Compute $(f^{-1})'(11)$.

Ex 9.1.28 Let $f(x) = ax+b$ with $a\neq 0$. Compute $(f^{-1})'(b)$. Why do we need the condition $a\neq 0$?

Ex 9.1.29 Let $f(x) =ax^2 +bx + c$ with $b\neq 0$. Compute $(f^{-1})'(c)$. Why do we need the condition $b\neq 0$?

Ex 9.1.30 Let $f(x) =a_n x^n + a_{n-1} x^{n-1 } + \cdots + a_1 x +a_0 =\sum_{k=0}^n a_k x^k$ with $a_1 \neq 0$. Compute $(f^{-1} )'(a_0)$. Why do we need the condition $a_1 \neq 0$?

Ex 9.1.31 Suppose that $f$ is injective on some interval containing $3$. If $f(3)=4$ and $f'(3)=6$ what is $(f^{-1})'(4)$?