# Functions and Relations

## Introduction

An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input.

## What is a Function?

Some relationships make sense and others don’t. Functions are relationships that make sense. All functions are relations, but not all relations are functions. A function is a relation that for each input, there is only one output.

Here are mappings of functions. The domain is the input or the x-value, and the range is the output, or the y-value. Each x-value is related to only one y-value. Athough the inputs equal to -1 and 1 have the same output, this relation is still a function because each input has just one output.

This mapping is not a function. The input for -2 has more than one output. ## Graphing Functions

Using inputs and outputs listed in tables, maps, and lists, makes it is easy to plot points on a coordinate grid. Using a graph of the data points, you can determine if a relation is a function by using the vertical line test. If you can draw a vertical line through a graph and touch only one point, the relation is a function.

Take a look at the graph of this relation map. If you were to draw a vertical line through each of the points on the graph, each line would touch at only one point, so this relation is a function.  ## Special Functions

Special functions and their equations have recognizable characteristics.

### Constant Function

$f(x) = c$

The c-value can be any number, so the graph of a constant function is a horizontal line. Here is the graph of $f(x) = 4$ ### Identity Function

$f(x) = x$

For the identity function, the x-value is the same as the y-value. The graph is a diagonal line going through the origin. ### Linear Function

$f(x) = mx + b$

An equation written in the slope-intercept form is the equation of a linear function, and the graph of the function is a straight line.

Here is the graph of $f(x)= 3x +4$ ### Absolute Value Function

$f(x) = |x|$

The absolute value function is easy to recognize with its V-shaped graph. The graph is in two pieces and is one of the piecewise functions. This is just a sample of the most common special functions.

## Inverse Functions

An inverse function reverses the inputs with its outputs.

$f(x) = 3x - 4$

Change the inputs with the outputs to create the inverse of this function.

\begin{align} f(x) &= 3x -4\\ y &= 3x -4\\ x &= 3y -4\\ x +4 &= 3y -4 + 4\\ x+ 4&= 3y\\ \frac{x + 4}{3}&= \frac{3}{3}y\\ f^{-1}(x)&=\frac{x + 4}{3} \end{align}

The inverse of $f(x) = 3x - 4$ is $f^{-1}(x) =\frac{x + 4}{3}$.

Not every inverse of a function is a function, so use the vertical line test to check.

## Function Operations

You can add, subtract, mutiply, and divide functions.

• $f(x) + g(x) = (f + g)(x)$
• $f(x) - g(x) = (f - g)(x)$
• $f(x) \times g(x) = (f \times g)(x)$
• $\frac{f(x)}{g(x)}= \frac{f}{g}(x)$

Look at two examples of function operations:

What is the sum of these two functions? Simply add the expressions.

\begin{align} f(x) &= 2x + 3\\ g(x) &= 3x + 5\\ (f + g) (x) &= 2x + 3 + 3x + 5 = 5x + 8 \end{align}

What is the product of these two functions? Simply multiply the expressions.

\begin{align} f(x) &= x + 4\\ g(x) &= x + 7\\ (f\times g)(x) &= (x + 4) \times (x +7) = x^{2} + 11x + 28 \end{align}