Inverse Functions

If the ordered pairs of a relation R are reversed, then the new set of ordered pairs is called the inverse relation of the original relation.

Example 1

If R = {(1,2), (3,8), (5,6)}, find the inverse relation of R. (The inverse relation of R is written R –1).

R –1 = {(2,1), (8,3), (6,5)}

Notice that the domain of R –1 is the range of R, and the range of R –1 is the domain of R. If a relation and its inverse are graphed, they will be symmetrical about the line y = x.

Example 2

Graph R and R –1 from Example along with the line y = x on the same set of coordinate axes.

The answer is shown in Figure 1.

If this graph were “folded over” the line y = x, the set of points called R would coincide with the set of points called R –1, making the two sets symmetrical about the line y = x.

  • Identity function. The function y = x, or f (x) = x, is called the identity function, since for each replacement of x, the result is identical to x.

  • Inverse function. Two functions, f and g, are inverses of each other when the composition f [ g( x)] and g[ f ( x)] are both the identity function. That is, f [ g( x)] = g[ f ( x)] = x.

Figure 1. Symmetrical sets of points.

figure

Example 3

If f ( x) = 4 x – 5, find f –1( x).

f ( x) = 4 x – 5 means  y = 4 x – 5

To find f –1( x), simply reverse the x and y variables and solve for y.

equation

For any ordered pair that makes f ( x) = 4 x – 5 true, the reverse ordered pair will make equation true.

To show that f ( x) and f –1( x) are truly inverses, show that their compositions both equal the identity function.

equation

Since f [ f –1( x)] = f –1[ f ( x)] = x, then f ( x) and f –1( x) are inverses of each other.

Example 4

Graph f ( x) and f –1( x) from Example together with the identity function on the same set of coordinate axes. The answer is shown in Figure 2.

Notice that if the graph were “folded over” the identity function, the graphs of f ( x) and f –1( x) would coincide.

Figure 2. Symmetrical graphs.

figure

Example 5

If f ( x) = x 2, find f –1( x).

  • f ( x) = x 2 means  y = x 2

equation

There are two relations for f –1( x), equation

In order for both f ( x) and f –1( x) to be functions, a restriction needs to be made on the domain of f ( x) so only one relation appears as f –1( x). If the domain of f ( x) is restricted to { x|x ≥ 0}, equation is the only answer for f –1( x). If the domain of f ( x) is restricted to { x|x ≤ 0}, then equation is the only answer for f –1( x).

Example 6

Graph f ( x) = x 2 together with equation, equation, and the identity function f (x) = x all on the same set of coordinate axes.

To graph f ( x) = x 2, find several ordered pairs that make the sentence y = x 2 true. To graph equation, simply take the reverse of the ordered pairs found for f ( x) = x 2. The graph is as shown in Figure 3.

x

f( x) = x 2

-3

9

-2

4

-1

1

0

0

1

1

2

4

3

9

x

equation

9

3

4

2

1

1

0

0

x

equation

9

-3

4

-2

1

-1

0

0

Notice that f ( x) = x 2 is a function but that equation is not a function. The reason is that equation does not pass the vertical line test. Also notice that f ( x) and f –1( x) will coincide when the graph is “folded over” the identity function. Thus, the two relations are inverses of each other.

Figure 3. f –1( x) is not a function.

figure

Example 7

Graph f ( x) = x 2 with the restricted domain { x| x ≥ 0} together with equation and the identity function on the same set of coordinate axes. The answer is shown in Figure 4.

Notice that f (x) and f –1( x) are now both functions, and they are symmetrical with respect to f ( x) = x. To show that f ( x) = x 2 and equation are inverse functions, show that their compositions each produce the identity function.

equation

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Figure 4. Solution to Example

figure

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