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# Points, Vectors, and Functions

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# Even and Odd Functions

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.