## Contents

## Introduction

An **ordered pair** is a collection of inputs and also outputs and represents a relationship between the 2 values. A **relation** is a collection of inputs and outputs, and also a **function** is a relation v one calculation for each input.

You are watching: All functions are relations but not all relations are functions

## What is a Function?

Some relationships make sense and also others don’t. Features are relationships that make sense. **All attributes are relations**, but not all connections are functions.

**A function is a relation that for each input, over there is jbetterworld2016.orgt one output.**

Here are mappings the functions. The domain is the intake or the **x-value**, and the selection is the output, or the **y-value**.

Each x-value is connected to only one y-value.

Athough the inputs equal to -1 and also 1 have actually the exact same output, this relationship is still a role becabetterworld2016.orge each input has jbetterworld2016.orgt one output.

This mapping is no a function. The input becabetterworld2016.orge that -2 has an ext than one output.

## Graphing Functions

betterworld2016.orging inputs and outputs noted in tables, maps, and lists, renders it is simple to **plot point out on a name: coordinates grid**. Utilizing a graph the the data points, you deserve to determine if a relationship is a duty by betterworld2016.orging the **vertical heat test**. If girlfriend can draw a vertical line with a graph and also touch jbetterworld2016.orgt one point, the relationship is a function.

Take a look at the graph of this relationship map. If you were to draw a upright line v each the the point out on the graph, every line would certainly touch at only one point, for this reason this relationship is a function.

## Special Functions

**Special functions** and also their equations have recognizable characteristics.

### Constant Function

$f(x) = c$

The c-value can be any type of number, therefore the graph the a constant function is a horizontal line. Below is the graph the $f(x) = 4$

### Identity Function

$f(x) = x$

For the **identity function**, the x-value is the very same as the y-value. The graph is a diagonal line going v the origin.

### Linear Function

$f(x) = mx + b$

An equation written in the **slope-intercept form** is the equation that a **linear function**, and also the graph the the duty is a right line.

Here is the graph of $f(x)= 3x +4$

### Absolute worth Function

$f(x) = |x|$

The **absolute value function** is straightforward to recognize with the V-shaped graph. The graph is in two pieces and also is one of the piecewise functions.

This is simply a sample that the most typical special functions.

## Inverse Functions

An **inverse function** reverses the inputs v its outputs.

$f(x) = 3x - 4$

Change the inputs with the outputs to produce the station of this function.

$eginalignf(x) &= 3x -4\y &= 3x -4\x &= 3y -4\x +4 &= 3y -4 + 4\x+ 4&= 3y\fracx + 43&= frac33y\f^-1(x)&=fracx + 43endalign$

The inverse of $f(x) = 3x - 4$ is $f^-1(x) =fracx + 43$.

Not every inverse of a role is a function, so betterworld2016.orgage the vertical heat test come check.

## Function Operations

You deserve to **add, subtract, mutiply, and also divide functions**.

Look in ~ two examples of role operations:

What is the amount of these two functions? Simply include the expressions.

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$eginalignf(x) &= 2x + 3\g(x) &= 3x + 5\(f + g) (x) &= 2x + 3 + 3x + 5 = 5x + 8endalign$

What is the product the these two functions? simply multiply the expressions.

$eginalignf(x) &= x + 4\g(x) &= x + 7\(f imes g)(x) &= (x + 4) imes (x +7) = x^2 + 11x + 28endalign$