SAT SUBJECT TEST MATH LEVEL 2

PART 2

REVIEW OF MAJOR TOPICS

CHAPTER 1
Functions

 
1.1 Overview

INVERSES

The inverse of a function f, denoted by –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 –1(x) = x.

To verify this, proceed as follows:

and

Since  is the inverse of f.

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

                      −1 ={(2,1), (3,2), (2,3)}

TIP 

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

To verify this, check  –1 and –1  term by term.

Thus, for each x, f(f –1(x)) = x.

Thus, for each x, f –1(f(x)) = x. In this case –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 about the line y = x.

3. –1 is not a function.

TIP 

Graphs of inverses are reflections about the line x.

4. –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 and and solve for y= 3+ 2, which becomes .

Thus,

6. f(x) = x2. Find f −1.

Write y = x2

Interchange x and yx = y2.

Solve for 

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

The inverse of any function can always be made a function by limiting the domain of f. In Example 6 the domain of could be limited to all nonnegative numbers or all nonpositive numbers. In this way –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 yx = 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) = , the inverse of f, f –1, could be represented by

           (A)  

           (B)  

           (C)  

           (D)  

           (E)  f –1 does not exist

3.       The inverse of = {(1,2),(2,3),(3,4),(4,1),(5,2)} would be a function if the domain of 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)  = 2x + 1

           (C)  

           (D)  

           (E)