## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 1

Functions

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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)}

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.

Graphs of inverses are reflections about the line |

**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*) = 3*x* + 2. Find *f* ^{−1}.

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

Thus,

**6.** *f*(*x*) = *x*^{2}. Find *f* ^{−1}.

Write *y* = *x*^{2}

Interchange *x* and *y*: *x* = *y*^{2}.

Solve for

Thus, the inverse of *y* = *x*^{2} 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*) = *x*^{2} and *x* ≥ 0. Find *f* ^{−1}.

Write *y* = *x*^{2} and *x* ≥ 0. Then switch *x* and *y*: *x* = *y*^{2} 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*) = 2*x* – 3, the inverse of *f, f *^{–1}, could be represented by

(A) *f* ^{–1}(*x*) = 3*x* − 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* = –2*x* + 1

(B) *y *= 2*x* + 1

(C)

(D)

(E)