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1.4 Inverse functions  (Page 2/11)

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Rule: horizontal line test

A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.

An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function “f(x) = x squared”, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function “f(x) = x cubed”, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.
(a) The function f ( x ) = x 2 is not one-to-one because it fails the horizontal line test. (b) The function f ( x ) = x 3 is one-to-one because it passes the horizontal line test.

Determining whether a function is one-to-one

For each of the following functions, use the horizontal line test to determine whether it is one-to-one.

  1. An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)
  2. An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function “f(x) = (1/x)”, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.
  1. Since the horizontal line y = n for any integer n ≥ 0 intersects the graph more than once, this function is not one-to-one.
    An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.
  2. Since every horizontal line intersects the graph once (at most), this function is one-to-one.
    An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function “f(x) = (1/x)”, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.
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Is the function f graphed in the following image one-to-one?

An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of the function “f(x) = (x cubed) - x” which is a curved function. The function increases, decreases, then increases again. The x intercepts are at the points (-1, 0), (0,0), and (1, 0). The y intercept is at the origin.

No.

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Finding a function’s inverse

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f to elements in the range of f . The inverse function maps each element from the range of f back to its corresponding element from the domain of f . Therefore, to find the inverse function of a one-to-one function f , given any y in the range of f , we need to determine which x in the domain of f satisfies f ( x ) = y . Since f is one-to-one, there is exactly one such value x . We can find that value x by solving the equation f ( x ) = y for x . Doing so, we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f . Consequently, this function is the inverse of f , and we write x = f −1 ( y ) . Since we typically use the variable x to denote the independent variable and y to denote the dependent variable, we often interchange the roles of x and y , and write y = f −1 ( x ) . Representing the inverse function in this way is also helpful later when we graph a function f and its inverse f −1 on the same axes.

Problem-solving strategy: finding an inverse function

  1. Solve the equation y = f ( x ) for x .
  2. Interchange the variables x and y and write y = f −1 ( x ) .

Finding an inverse function

Find the inverse for the function f ( x ) = 3 x − 4 . State the domain and range of the inverse function. Verify that f −1 ( f ( x ) ) = x .

Follow the steps outlined in the strategy.

Step 1. If y = 3 x − 4 , then 3 x = y + 4 and x = 1 3 y + 4 3 .

Step 2. Rewrite as y = 1 3 x + 4 3 and let y = f −1 ( x ) .

Therefore, f −1 ( x ) = 1 3 x + 4 3 .

Since the domain of f is ( − ∞ , ∞ ) , the range of f −1 is ( − ∞ , ∞ ) . Since the range of f is ( − ∞ , ∞ ) , the domain of f −1 is ( − ∞ , ∞ ) .

You can verify that f −1 ( f ( x ) ) = x by writing

f −1 ( f ( x ) ) = f −1 ( 3 x − 4 ) = 1 3 ( 3 x − 4 ) + 4 3 = x − 4 3 + 4 3 = x .

Note that for f −1 ( x ) to be the inverse of f ( x ) , both f −1 ( f ( x ) ) = x and f ( f −1 ( x ) ) = x for all x in the domain of the inside function.

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Find the inverse of the function f ( x ) = 3 x / ( x − 2 ) . State the domain and range of the inverse function.

f −1 ( x ) = 2 x x − 3 . The domain of f −1 is { x | x ≠ 3 } . The range of f −1 is { y | y ≠ 2 } .

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Graphing inverse functions

Let’s consider the relationship between the graph of a function f and the graph of its inverse. Consider the graph of f shown in [link] and a point ( a , b ) on the graph. Since b = f ( a ) , then f −1 ( b ) = a . Therefore, when we graph f −1 , the point ( b , a ) is on the graph. As a result, the graph of f −1 is a reflection of the graph of f about the line y = x .

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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