# Points, Vectors, and Functions

### Topics

## Introduction to Points, Vectors, And Functions - At A Glance:

Unlike the typical college junior that shows up to their morning class wearing pajamas and their retainer, some functions care what they look in the mirror. These functions, called **even** or **odd** functions, have some important properties we can take advantage of later.

Consider the *y*-axis to be a mirror. A function is *even* if it looks in the mirror and sees itself exactly as is. In other words, it looks the same when reflected across the *y*-axis.

For any value of *x*, the values *f*(*x*) and *f*(*-x*) must be the same.

In symbols, a function is even if *f*(*x*) = *f*(*-x*).

To check if a function is even we find *f*(*x*) and *f*(*-x*) and see if they're the same.

Alternatively, a function is *odd* if it looks in the mirror and sees itself standing upside-down, like the *y*-axis is a funhouse mirror.

For any value of *x*, *f*(*-x*) is the upside-down version of *f*(*x*). That is, *f*(*x*) and *f*(*-x*) are negatives of each other.

In symbols, a function is odd if
*f*(*-x*) = -*f*(*x*).

If all the terms involve *x* raised to even exponents, the polynomial is even. Some examples are

*f*(*x*) = *x*^{2}*f*(*x*) = 6*x*^{4}-7*x*^{2}*f*(*x*) = *x*^{8} + *x*^{6} + *x*^{22}

If all the exponents are odd, the polynomial is odd. Some examples are

*f*(*x*) = *x*^{3 }

*f*(*x*) = *x*^{9 }-6*x*^{7}

*f(**x) = x*^{101}* + x*^{67}

If some of the exponents are odd and some of the exponents are even, the polynomial is neither even nor odd.

#### Example 1

Is the function |

#### Example 2

Is the function |

#### Example 3

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#### Exercise 1

By comparing the values *f*(*x*) and *f*(*-x*), determine whether each function is (a) even or (b) odd. Use a graphing calculator to check your answer.

*f*(*x*) = 6

#### Exercise 2

By comparing the values *f*(*x*) and *f*(*-x*), determine whether each function is (a) even or (b) odd. Use a graphing calculator to check your answer.

*f*(*x*) = -x

#### Exercise 3

By comparing the values *f*(*x*) and *f*(*-x*), determine whether each function is (a) even or (b) odd. Use a graphing calculator to check your answer.

*f*(*x*) =*x*^{3}+*x*^{4}

#### Exercise 4

*f*(*x*) and *f*(*-x*), determine whether each function is (a) even or (b) odd. Use a graphing calculator to check your answer.

*f*(*x*) = 0