- Topics At a Glance
**Functions**- Increasing or Decreasing or...
- Bounded
**Even and Odd Functions**- Vectors: A New Kind of Animal
- Magnitude
- Direction
- Scaling Vectors
- Unit Vectors
- Vector Notation
- (More than 2)-Dimensional Vectors
- Vector Functions
- Sketching Vector Functions
- Parametric Equations
- Graphing Parametric Equations
- Points on Graphs of Parametric Equations
- Parametrizations of the Unit Circle
- Parameterization of Lines
- Polar Coordinates
- Simple Polar Inequalities
- Switching Coordinates
- Translating Equations and Inequalities between Coordinate Systems
- Polar Functions
- Graphing Polar Functions
- Rules of Graphing We Do (or Don't) Have
- Bounds on Theta
- Intersections of Polar Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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

Is the function |

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