﻿ ﻿ODD AND EVEN FUNCTIONS - Overview - Functions - REVIEW OF MAJOR TOPICS - SAT SUBJECT TEST MATH LEVEL 2

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

### 1.1 Overview ### ODD AND EVEN FUNCTIONS

A relation is said to be even if (–x,y) is in the relation whenever (x,y) is. If the relation is defined by an equation, it is even if (–x,y) satisfies the equation whenever (x,y) does. If the relation is a function f, it is even if f(–x) = f(x) for all x in the domain of f. The graph of an even relation or function is symmetric with respect to the y axis.

EXAMPLES

1. {(1,0),(–1,0),(3,0),(–3,0),(5,4),(–5,4)} is an even relation because (–x,y) is in the relation whenever (x,y) is.

2. x4 + y2 = 10 is an even relation because (–x)4 + y2 = x4 + y2 = 10.

 TIP An even relation is symmetric about the y-axis.

3. f(x) = x2 and f(–x) = (–x)2 = x2. 4. f(x) = | x | and f(−x) = | −x | = | −1 · x | = | −1 | · | x | = | x |. A relation is said to be odd if (–x,–y) is in the relation whenever (x,y) is. If the relation is defined by an equation, it is odd if (–x,–y) satisfies the equation whenever (x,y) does. If the relation is a function f, it is odd if f(–x) = –f(x) for all x in the domain of x. The graph of an odd relation or function is symmetric with respect to the origin.

5. {(5,3),(–5,–3),(2,1),(–2,–1),(–10,8), (10,–8)} is an odd relation because (–x,–y) is in the relation whenever (x,y) is.

6. x4 + y2 = 10 is an odd relation because (–x)4 + (–y)2 = x4 + y2 = 10. Note that x4 + y2 = 10 is both even and odd.

 TIP An odd relation is symmetric about the origin.

7. f(x) = x3 and f(–x) = (–x)3 = –x3.

Therefore, f(–x) = x3 = –f(x). TIP Relations can be either odd, even, or neither. They can also be both odd and even!

8. and .

Therefore, . The sum of even functions is even. The sum of odd functions is odd. The product of an even function and an odd function is odd. The product of two even functions or two odd functions is even.

EXERCISES

1. Which of the following relations are even?

I. y = 2

II. f(x) = x

III. x2 + y2 = 1

(A) only I

(B) only I and II

(C) only II and III

(D) only I and III

(E) I, II, and III

2. Which of the following relations are odd?

I. y = 2

II. y = x

III. x2 + y2 = 1

(A) only II

(B) only I and II

(C) only I and III

(D) only II and III

(E) I, II, and III

3. Which of the following relations are both odd and even?

I. x2 + y2 = 1

II. x2y2 = 0

III. x + y = 0

(A) only III

(B) only I and II

(C) only I and III

(D) only II and III

(E) I, II, and III

4. Which of the following functions is neither odd nor even?

(A) {(1,2),(4,7),(–1,2),(0,4),(–4,7)}

(B) {(1,2),(4,7),(–1,–2),(0,0),(–4,–7)}

(C) y = x3 – 1

(D) y = x2 – 1

(E) f(x) = –x

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