﻿ ﻿INVERSES - Overview - Functions - REVIEW OF MAJOR TOPICS - SAT SUBJECT TEST MATH LEVEL 2

## PART 2 ## REVIEW OF MAJOR TOPICS ## CHAPTER 1Functions

### 1.1 Overview ### INVERSES

The inverse of a function f, denoted by f –1, is a relation that has the property that f(x) f –1(x) = f –1(x) f(x) = x. The inverse of a function is not necessarily a function.

EXAMPLES

1. the inverse of f ?

To answer this question assume that and verify that f(x) f –1(x) = x.

To verify this, proceed as follows: and Since is the inverse of f.

2. f = {(1,2),(2,3),(3,2)}. Find the inverse.

f −1 ={(2,1), (3,2), (2,3)}

 TIP Remember that the inverse of a function need not be a function.

To verify this, check f f –1 and f –1 f term by term. Thus, for each x, f(f –1(x)) = x. Thus, for each x, f –1(f(x)) = x. In this case f –1 is not a function.

If the point with coordinates (a,b) belongs to a function f, then the point with coordinates (b,a) belongs to the inverse of f. Because this is true of a function and its inverse, the graph of the inverse is the reflection of the graph of f about the line y = x.

3. f –1 is not a function. TIP Graphs of inverses are reflections about the line y = x.

4. f –1 is a function. As can be seen from the above examples, the graph of an inverse is the reflection of the graph of a function (or relation) through the line y = x. Algebraically, the equation of an inverse of a function can be found by replacing f (x) by y; interchanging x and y; and solving the resulting equation for y.

5. f(x) = 3x + 2. Find f −1.

In order to find f –1, interchange x and y and solve for y: x = 3y + 2, which becomes .

Thus, 6. f(x) = x2. Find f −1.

Write y = x2

Interchange x and y: x = y2.

Solve for Thus, the inverse of y = x2 is not a function.

The inverse of any function f can always be made a function by limiting the domain of f. In Example 6 the domain of f could be limited to all nonnegative numbers or all nonpositive numbers. In this way f –1 would become either or , both of which are functions.

7. f(x) = x2 and x ≥ 0. Find f −1.

Write y = x2 and x ≥ 0. Then switch x and y: x = y2 and y ≥ 0.

Solve for y: .

Here f −1 is the function  Finding an equation for the inverse of a function can also be used to determine the range of a function, as shown in the following example.

8. Find the range of .

First replace f(x) by y, and interchange x and y to get . Then solve for y: In order for this to be defined, x ≠ −2. In other words, −2 is not in the range of f. (This could also be determined by observing that in the original function can never be zero.)

EXERCISES

1. If f(x) = 2x – 3, the inverse of f, f –1, could be represented by

(A) f –1(x) = 3x − 2

(B) (C) (D) (E) 2. If f(x) = x , the inverse of f, f –1, could be represented by

(A) (B) (C) (D) (E) f –1 does not exist

3. The inverse of f = {(1,2),(2,3),(3,4),(4,1),(5,2)} would be a function if the domain of f is limited to

(A) {1,3,5}

(B) {1,2,3,4}

(C) {1,5}

(D) {1,2,4,5}

(E) {1,2,3,4,5}

4. Which of the following could represent the equation of the inverse of the graph in the figure? (A) y = –2x + 1

(B) y = 2x + 1

(C) (D) (E) ﻿