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Topic: Inverse Functions

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Domain and Range | One-to-One Functions | Inverse Functions | Graphing Inverse Functions | Finding Inverse Functions Algebraically | Logarithmic and Exponential Functions

Domain and Range

It sometimes happens that one function "undoes" what another function does. For instance, if $f(x) = 2x$  and $g(x) = x/2,$ then $f$ doubles input numbers, while $g$ does the opposite. We shall refer to $f$ and $g$ as inverse functions. But before we can discuss inverse functions properly, we shall first need to review what is meant by the domain of a function, discuss the range of a function, and look at the manner in which functions interact with each other.

First, we review the concept of "domain" from the book.

Domain of a Function
If $f$ is a function, then the domain of $f$ is the set of all real numbers $x$ for which $f(x)$ is defined. Sometimes the domain of a function is specified explicitly. If no domain is specified for the function $f,$ we take the domain to be the largest set of numbers x for which $f(x)$ makes sense. This "largest possible domain" is sometimes called the natural domain.

Quick Examples
1. Let $N(t)$ be the total number of CD burners your electronics factory has manufactured in $t$ months since it began operatiing. The factory has a maximum capacity of $1000$ per month.

    An appropriate domain for $N$ is

2. Let $f(x) = (x - 2)^{0.5}$

    The natural domain of $f$ is

3.Let $g(x) = \frac{4}{x + 5}$

    The natural domain of $g$ is

Related to the above concept is the concept of the "range" of a function.

Range of a Function
If $f$ is a function, then the range of $f$ is the set of all possible values for $f(x).$

If we think of a function as machine -- in goes $x,$ out goes $f(x)$ -- then the range of $f$ is the set of all possible numbers that it spews out.

Quick Examples
1. The range of the function $f$ given by $f(x) = 2x$ consists of all possible values of $f(x),$ such as $f(1) = 2,$ $f(1/2) = 1,$ $f(-3) = -6,$ and so on. In fact, we can get any number we please this way. For example, to get $-600,$ just take $f(-300).$ More precisely, to get an arbitrary value $y,$ we just take $f(y/2).$ Thus the range is the set of all real numbers.

2. Let $f(x) = x^2.$   Trying a few values first, $f(0) = 0,$ $f(1) = 1,$ $f(-1) = 1,$...   It appears that we can only get non-negative numbers this way. In fact, we can get any nonnegative number $y$ we like: if $y ≥ 0,$ then $f(y^{0.5}) = (y^{0.5})^2 = y.$ Thus the range of $f$ is $[0, +∞)$ -- the set of all non-negative real numbers.

3. Let $f(x) = x^2 + 1.$

    The range of $f$ is

Here is a graphical way of obtaining the range of a function. As an example, let us use $f(x) = x^2,$ whose graph is shown below. The range of $f$ consists of all the possible $y$-values we can get. This amounts to finding all the heights of points on the graph of $f.$ To find this graphically, pretend that a pair of elevator doors come in from both sides and squash the graph onto the $y$-axis. Then the resulting "squashed" graph gives the range of the function.

Looking at the "squashed" graph, we find it covers the interval $[0, +∞)$ on the $y$-axis, confirming that the range is $[0, +∞).$

Example 1 Finding the Range of a Function

Find the range of the function $g(x) = x + 1/x.$

Instead of attempting to do this algebraically, we use the elevator door method, illustrated above. Here is the graph of $g.$ Try to imagine what the graph will look like when squashed onto the $y$-axis before pressing the button!

Before we go on... Although graphers don't come equipped with elevator doors to squash graphs for us, you can use one to approximate the squashing effect as follows: Try plotting the graph of $g(x) = x + 1/x$ with the following $x$-range: $0.1 ≤ x ≤ 100,$ but keeping the $y$-range quite small-e.g. $-5 ≤$ y $≤ 5.$ The effect will be as though the graph has been squeezed close to the y-axis, showing the range of the function quite clearly (assuming that the calculator draws it accurately enough). (You could try it on the Excel Grapher or one of the other graphers listed at the top of the page.) It may not be a good idea to try the range $-100 ≤ x ≤ 100,$ since most graphers will try to connect the two pieces of the graph.

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One-to-One Functions

We have seen in the Section 1.2 in the text-book that a graph in the $xy-$plane is the graph of a function with independent variable $x$ if it passes the "vertical line test." That is, if each vertical line passes through at most one point of the graph.

Now pretend that we have the graph of some function that passes the "horizontal line test." In other words, suppose that each horizontal line passes through at most one point of the graph. The next figure shows the graphs of some functions, one which passes the horizontal line test, and one which fails it. Those that pass it are injective. (For an algebraic definition, see below.)

Each horizontal line passes through at most one point of the graph.
Not one-to-one
Some horizontal linea pass through more than one point of the graph.
Not one-to-one
Not one-to-one
Not one-to-one

To interpret what it means algebraically for a function to be one-to-one, take a look at the graph of a function that is not one-to-one.

First, notice that the horizontal line through $y = 2$ passes through two points: $(-4, 2)$ and $(0, 2).$ Another way of saying this is that $f(-4) = f(0) = 2.$  Similarly, $f(1) =f(3) = 0,$  so that again we have different $x$-values that give the same value for $f(x).$  Also, $f(-1) = f(-3) = 4.$

This does not happen in the case of one-to-one functions; it cannot happen that $f(a) = f(b)$  for two different $x$-values $a$ and $b.$  It is for this reason that we call such a function one-to-one. (One-to-one functions are also called injective.)

One-to-One Function
A function $f$ is one-to-one if it never happens that $a ≠ b,$ but $f(a) = f(b).$  Equivalently, one-to-one functions are those functions whose graphs pass the horizontal line test: every horizontal line passes through at most one point of its graph.

Example 2 Determining Whether a Function is One-to-One

Which of the following functions is one-to-one?

All we need to do is sketch their graphs and decide which of them pass the horizontal line test.
(a) The graph of $f(x) = 4 - x^2$  fails the horizontal line test, since the line $y = 0$ passes through the two points $(-2, 0)$ and $(2, 0).$ In other words, f(-2) = f(2) = 0, so this function is not one-to-one.

(b) The function $g(x) = x^3$ is One-to-one
Not one-to-one
(c)The function $h(x) = 4 - x^2$ with domain $[0, + ∞)$ is One-to-one
Not one-to-one

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Inverse Functions

Suppose you take a number $x,$ double it and then halve the answer. You expect to get $x$ back again. In a sense, halving a number is the reverse process of doubling it. Similarly, if you start with $x,$ cube it, and then take the cube root of the answer, you again get back $x.$ Thus taking the cube root of a number is the reverse process of cubing it. Let us express these simple situations in terms of functions.

Example 3 Composing Inverse Functions

Let $f(x) = 2x$  (the doubling function), and let $g(x) = x/2$ (the halving function). Calculate $g(f(x))$ and $f(g(x)).$

To evaluate expressions such as these, we start from the inside and work outward:


Thus, $g(f(x)) = x$ and $f(g(x)) = x.$ This is how we say that the halving function and the doubling function are the "reverse" of each other. Formally, we refer to $f$ and $g$ as being inverse functions.

Before we go on...
Here is a nice way of visualizing what is going on. Since $g(f(x))$ means "first apply $f$ and then apply $g,$" we can think of this as feeding the output of $f$ into $g,$ and seeing what we get. Here is an illustration of this process.

See if you can illustrate the corresponding process for $f(g(x)).$

Inverse Functions
The functions $f$ and $g$ are called inverse functions if:

    $g(f(x)) = x$  for every $x$ in the domain of $f,$ and
    $f(g(x)) = x$  for every $x$ in the domain of $g.$

When $f$ and $g$ are inverse functions, we write $g(x)$ as $f^{-1}(x).$

For $g(f(x))$ to be defined, it must also be the case that $f(x)$ is in the domain of $g$ for every $x$ in the domain of $f.$ Similarly, for $f(g(x))$ to be defined, it must be the case that $g(x)$ is in the domain of $f$ for every $x$ in the domain of $g.$

1. We saw above that $f(x) = 2x$ and $g(x) = x/2$ are inverse functions. Thus, $f^{-1}(x) = g(x) = x/2.$

2. Let $f(x) = x^3 + 1$ and $g(x) = (x - 1)^{1/3}.$ Then
    $g(\color{blue}{f(x)}) = g(\color{blue}{x^3 +1}) = [\color{blue}{(x^3 +1)} -1] ^{1/3} = (x^3)^{1/3} = x$

$f(g(x))$ $=$
$f($ $)$

3. Let $f(x) = x^2 + 1,$ with domain $[0, +∞)$ -- the set of all non-negative real numbers. The inverse of $f$ is (Check by computing $f(g(x))$ and $g(f(x)).$   

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Graphing Inverse Functions

Q How are the graphs of a function and its inverse related?
A Take a look at the figure which shows the graphs of $f(x) = x^2+1$ and its inverse, $f^{-1}(x) = (x - 1)^{1/2}.$

We have also included the graph of $y = x$ to show the symmetry. Pretend that you held the two ends of the line $y = x$ and flipped the portion of the coordinate plane shown so that the $x$-axis wound up on top of the original $y$-axis. Then the graph of f would land exactly on top of the graph of $f^{-1}.$ In other words:

Also notice something else from the graph: The range of $f$ is $[2, +∞)$ and the domain of $f^{-1}$ is also $[2, +∞).$ This is true in general:

(Note that, for example, $(2, 5)$ is a point on the graph of $f,$ and the corresponding point on the graph of $f^{-1}$ is $(5, 2).$ This is the way that individual points on the two graphs correspond: we interchange coordinates to go from the graph of $f$ to the graph of $f^{-1}.$

Q Why is this so?
A Let's suppose that the inverse functions we are considering are called $f$ and $f^{-1}.$  Then their graphs are the graphs of the equations $y = f(x)$  and $y = f^{-1}(x).$  Look at the equation $y = f(x)$ for a moment. If we apply $f^{-1}$ to both sides, we get $f^{-1}(y) = f^{-1}(f(x)) = x,$    in other words, $x = f^{-1}(y).$  Thus, a point on the graph of $f$ also satisfies the equation $x = f^{-1}(y).$  In other words, it is a point on the graph of $f^{-1}$ if the $x$- and $y$-axes are interchanged. This means that the graph of $f$ is the same as the graph of $f^{-1},$ but with the roles of $x$ and $y$ interchanged. An easy way to interchange $x$ and $y$ is to flip the coordinate plane about the line $y = x$ as shown below.

Q Do all functions have inverses?
A No. Think of how the graph of a function is related to the graph of its inverse. If the inverse function is to exist, then its graph is obtained from the graph of the original function by flipping about the line $y = x.$ Now the resulting graph had better be the graph of a function, so it had better pass the vertical line test. But vertical lines correspond to horizontal lines under the operation of flipping. Thus, in order for the graph of the inverse to pass the vertical line test, it had better be the case that the function we started with passed the horizontal line test. This amounts to saying that the original function had better be one-to-one if it is to have an inverse!

Q Do all one-to-one functions have inverses?
A Yes. If $f$ is one-to-one, then the flipping operation results in a graph that passes the vertical line test, and so is the graph of a function. A little thought will convince you that this function undoes what the function $f$ did-in other words, that the new function is the inverse of $f.$

Here is a summary of what we have learned so far.

Graphing the Inverse of a One-to-One Function
  1. If $f$ is a one-to-one function, then it has an inverse $f^{-1}.$  If $f$ is not one-to-one, then it does not have an inverse.
  2. The domain of $f^{-1}$ is the same as the range of $f;$ the range of $f^{-1}$ is the same as the domain of $f.$
  3. To obtain the graph of $f^{-1}$ from that of $f,$ flip the graph of $f$ about the line $y = x,$ so that the $x$-axis is superimposed on the old $y$-axis and vice-versa. The graph obtained is then the graph of $f^{-1}.$
  4. Points on the graph of $f^{-1}$ are obtained from corresponding points on the graph of $f$ by switching the $x$- and $y$-coordinates.


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Finding Inverse Functions Algebraically

Q I can obtain the graph of $f^{-1}$ by flipping the graph of the one-to-one function $f$ about the line $y = x.$ This is a graphical way of obtaining the inverse. Is there an algebraic way of doing this? In other words, if I am given a formula for $f(x),$ how do I get the formula for $f^{-1}(x)$?
A This is answered by the next example.

Example 4 Finding the Inverse of a Function

Find the inverse of the function $f(x) = x^{1/3} + 2$

First, we must check that this function is one-to-one. Its graph is the graph of $y = x^{1/3}$ shifted up $2$ units. To convince yourself that it passes the horizontal line test, either graph it on the Function Evaluator & Grapher or Excel Grapher. Alternatively, press here to see a picture of the graph. Thus the function f is one-to-one, and hence does have an inverse.

Here is a two-step method to find the inverse:
  1. Write $y = f(x)$  and solve for $x.$
    $y = x^{1/3} + 2$ gives:
    $x^{1/3} = y - 2,$
    whence $x = (y - 2)^3.$
  2. Switch $x$ and $y$ in the resulting equation. The resulting function of $x$ is the inverse.
    Switching $x$ and $y$ gives
    $y = (x - 2)^3.$
    Thus the inverse function is
    $f^{-1}(x) = (x - 2)^3.$

Before we go on... This process yields the inverse because of the above discussion on graphing an inverse function: the equation associated with the inverse function of a given function $f$ is the curve $y = f(x),$ but with $x$ and $y$ reversed. Thus, to see the form of this equation, we must write it in the form $x = g(y),$ which explains why we solved for $x.$

This process won't work for a function that is not one-to-one. For example, when we try to solve $y = x^2 + 1$ for $x,$ we get two possible solutions; $x =(y-1)^{1/2}$ and $x = -(y-1)^{1/2}.$ In this case, the given function $f$ is not invertible; if you look at its graph (a parabola) and flip it about the line $y = x,$ you get a curve (a sideways parabola) that is not the graph of a function, as it fails the vertical line test. Alternatively, the graph of $f$ fails the horizontal line test.

Here is one for you to participate in.

Example 5 Finding the Inverse of a Function

Find the inverse of the function

with domain $(-1, +∞).$

As a first step, replace $"f(x)"$  by $y,$ to obtain Now solve the equation for $x:$

Finally, replace $y$ by $x$ to obtain the formula for the inverse:

Here are the graphs of $f$ and its inverse.

Graph of $f$ Graph of $f^{-1}$ Both graphs

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Logarithmic and Exponential Functions

Here is the graph of the exponential function $f(x) = 2^x.$

To obtain its inverse, we set

and solve for $x.$ But doing this amounts to nothing more than writing the above equation in logarithmic form:

Hence the inverse of the exponential function $f(x) = 2^x$ is

Here are the graphs of $f$ and $f^{-1}$ on the same set of axes.

The relationship between logarithmic and exponential functions does not, or course, depend on the base 2, and we have the following more general result.

Exponential and Logarithmic Functions
  • The functions $f(x) = b^x$ and $g(x) = \log_bx$ are inverse functions.
  • The domain of $f(x) = b^x$ is te set of all real numbers, and the domain of $g(x) = \log_bx$ is $(0, +∞)$

The following identities follow from the inverse relationship between $f$ and $g.$
    $\log_b( b^x) = x$$g(f(x)) = x$
    $b^{\log_bx} = x$$f(g(x)) = x$

In particular, we can choose the base $b$ to be either $e$ or $10$ and obtain:
    $\ln( e^x) = x$Base $e$
    $e^{\ln x} = x$Base $e$
    $\log( 10^x) = x$Base $10$
    $10^{\log x} = x$Base $10$

Q Is there a direct way of seeing why those identities are true?
A Look at the first identity,

The left-hand side, $\log_b(b^x),$ means the power to which you must raise $b$ in order to get $b^x,$ and this is obviously the $x$ power. Turning to the second identity,

the left-hand side is, in words, $b$ raised to the power to which you must raise $b$ in order to get $x.$ Surely that must give you $x$! (It's like saying "The name of the person whose name is Earl, is Earl!")

You can now now go on and try the exercise set for this topic.

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Last Updated:February, 2003

Copyright © 2001 StefanWaner and Steven R. Costenoble