### Learning Outcomes

- Divide two fractions
- Divide a fraction by a whole number
- Determine whether a function is even, odd, or neither

In the Integration Formulas and the Net Change Theorem section, we will need skills that include how to simplify complex fractions and determine the symmetry of a function. These skills are reviewed here.

## Simplify Complex Fractions

When integrating, it is not uncommon to have to work with a complex fraction, that is, a fraction that contains fractions in its numerator/denominator. An easy way to remember how to divide fractions is to multiply by the reciprocal. This means that you keep the first number as-is, change the division sign to multiplication, and then find the reciprocal of the second number (flip it!).

### Example: Simplifying Complex Fractions

Simplify [latex]\dfrac{\frac{2}{3}}{\frac{1}{6}}[/latex].

### Example: Simplifying Complex Fractions

Divide [latex]\dfrac{\frac{3}{5}}{2}[/latex].

### Try It

Simplify [latex]\dfrac{\frac{5}{7}}{\frac{3}{8}}[/latex].

## Determine Whether a Functions is Even, Odd, or Neither

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\left(x\right)={x}^{2}[/latex] or [latex]f\left(x\right)=|x|[/latex] will result in the original graph. We say that these types of graphs are symmetric about the [latex]y[/latex]-axis. Functions whose graphs are symmetric about the *y*-axis are called **even functions.**

If the graphs of [latex]f\left(x\right)={x}^{3}[/latex] or [latex]f\left(x\right)=\dfrac{1}{x}[/latex] were reflected over *both* axes, the result would be the original graph.

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an **odd function**.

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\left(x\right)={2}^{x}[/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\left(x\right)=0[/latex].

You can view the transcript for “Introduction to Odd and Even Functions” here (opens in new window).

### A General Note: Even and Odd Functions

A function is called an **even function** if for every input [latex]x[/latex]

[latex]f\left(x\right)=f\left(-x\right)[/latex]

The graph of an even function is symmetric about the [latex]y\text{-}[/latex] axis.

A function is called an **odd function** if for every input [latex]x[/latex]

[latex]f\left(x\right)=-f\left(-x\right)[/latex]

The graph of an odd function is symmetric about the origin.

### How To: Given the formula for a function, determine if the function is even, odd, or neither.

- Determine whether the function satisfies [latex]f\left(x\right)=f\left(-x\right)[/latex]. If it does, it is even.
- Determine whether the function satisfies [latex]f\left(x\right)=-f\left(-x\right)[/latex]. If it does, it is odd.
- If the function does not satisfy either rule, it is neither even nor odd.

### Example: Determining whether a Function Is Even, Odd, or Neither

Is the function [latex]f\left(x\right)={x}^{3}+2x[/latex] even, odd, or neither?

### Try It

Is the function [latex]f\left(s\right)={s}^{4}+3{s}^{2}+7[/latex] even, odd, or neither?