How do you find the inverse of a relation

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.

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.

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

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

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.

Example 5

If f ( x) = x 2, find f –1( x).

• f ( x) = x 2 means y = x 2

There are two relations for f –1( x),

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},

Example 6

Graph f ( x) = x 2 together with

To graph f ( x) = x 2, find several ordered pairs that make the sentence y = x 2 true. To graph

x
9	3
4	2
1	1
0	0

x
9	-3
4	-2
1	-1
0	0

Notice that f ( x) = x 2 is a function but that

Figure 3. f –1( x) is not a function.

Example 7

Graph f ( x) = x 2 with the restricted domain { x| x ≥ 0} together with

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

Figure 4. Solution to Example