# Function

How to determine whether a relation is a function and how to find the function value: 5 examples and their solutions.

## Example

A function is a relation

that shows [one x, one y].

Let's see how to determine

whether the given relation is a function or not.

Draw two sets.

Fill the x values in the left set: 1, 2, 3, 4.

(The left set, x, is called the domain.)

Fill the y values in the right set: 2, 1, 0, 3.

(The right set, y, is called the range.)

Connect each x and the paired y.

1 is paired with 2.

So connect the left 1 and the right 2.

2 is paired with 1.

So connect the left 2 and the right 1.

3 is paired with 0.

So connect the left 3 and the right 0.

4 is paired with 3.

So connect the left 4 and the right 3.

Each left number shows [one x, one y].

So this relation is a function.

## Example

Draw two sets.

Fill the x values in the left set: 1, 2, 3, 4.

Fill the y values in the right set: 2, 4, 1.

Connect each x and the paired y.

1 is paired with 2.

So connect the left 1 and the right 2.

2 is also paired with 2.

So connect the left 2 and the right 2.

3 is paired with 4.

So connect the left 3 and the right 4.

4 is paired with 1.

So connect the left 4 and the right 1.

Each left number shows [one x, one y].

So this relation is a function.

## Example

Draw two sets.

Fill the x values in the left set: 1, 2, 3, 4.

Fill the y values in the right set: 3, 1, 4, 2.

Connect each x and the paired y.

1 is paired with 3.

So connect the left 1 and the right 3.

2 is paired with 1 and 4.

So connect the left 2 and the right 1.

And connect the left 2 and the right 4.

3 is paired with 4.

So connect the left 3 and the right 4.

4 is paired with 2.

So connect the left 4 and the right 2.

The left 2 shows [one x, two y]:

2 is paired with 1 and 4.

This is not [one x, one y].

So this relation is not a function.

## Examplef(x) = 3x - 1, f(-2) = ?

f(x) is the y of a function if x = x.

f(x) is read as [f x] or [f of x].

In this example,

if x = x, then y = 3x - 1.

So f(-2) = -7.

This means

if x = -2, then y = -7.

To find f(-2),

put -2 into f(x) = 3x - 1.

Then f(-2) = 3⋅(-2) - 1.

Then 3⋅(-2) - 1 = -7.

So f(-2) = -7.

This means

if x = -2, then y = -7.

## Examplef(x) = 2x + 5, f(k) = 13, k = ?

So k = 4 is the answer.

This means

if x = 4, then y = 13.

To find f(k),

put k into f(x) = 2x + 5.

Then f(k) = 2k + 5.

It says f(k) = 13.

And f(k) = 2k + 5.

So 2k + 5 = 13.

Solve 2k + 5 = 13.

Then k = 4.

Linear Equation (One Variable)

So k = 4 is the answer.

This means

if x = 4, then y = 13.