# Function

How to solve function problems: meaning of *y* = *f*(*x*), examples, and their solutions.

## Example 1

A function is a relation

that shows [one input], [one output].

In this example,*x* is the input

and *y* is the output.

The input, *x*, is also called as the [domain].

And the output, *y*, is also called as the [range].

To see if this relation is a function or not:

Draw two groups like this.

Write the *x* values in the left group:

1, 2, 3, 4.

Write the *y* values in the right group:

2, 1, 0, 3.

1 is paired with 2.

So draw an arrow from 1 to 2.

2 is paired with 1.

So draw an arrow from 2 to 1.

3 is paired with 0.

So draw an arrow from 3 to 0.

4 is paired with 3.

So draw an arrow from 4 to 3.

[Each number] in the left group

has [one output] (its paired number) in the right group.

This shows [one input], [one output].

So this relation is a function.

## Example 2

Draw two groups like this.

Write the *x* values in the left group:

1, 2, 3, 4.

Write the *y* values in the right group:

3, 2, 0.

1 is paired with 3.

So draw an arrow from 1 to 3.

2 is not paired with any number.

So don't draw an arrow.

3 is paired with 2.

So draw an arrow from 3 to 2.

4 is paired with 0.

So draw an arrow from 4 to 0.

2 in the left group

doesn't have any output.

This shows [one input], [no output].

So this relation is not a function.

## Example 3

Draw two groups like this.

Write the *x* values in the left group:

1, 2, 3, 4.

Write the *y* values in the right group:

2, 4, 1.

1 is paired with 2.

So draw an arrow from 1 to 2.

2 is also paired with 2.

So draw an arrow from 2 to 2.

3 is paired with 4.

So draw an arrow from 3 to 4.

4 is paired with 1.

So draw an arrow from 4 to 1.

[Each number] in the left group

has [one output] in the right group.

It doesn't matter

whether 1 and 2 have the same output: 2.

If each number in the left group has one output,

then it's okay.

So the relation shows [one input], [one output].

So this relation is a function.

## Example 4

Draw two groups like this.

Write the *x* values in the left group:

1, 2, 3, 4.

Write the *y* values in the right group:

3, 1, 4, 2.

1 is paired with 3.

So draw an arrow from 1 to 3.

2 is paired with 1 and 4.

So draw two arrows from 2 to 1 and 4.

3 is paired with 4.

So draw an arrow from 3 to 4.

4 is paired with 2.

So draw an arrow from 4 to 2.

2 in the left group

has two outputs: 1 and 4.

This shows [one input], [two outputs].

So this relation is not a function.

## Meaning of *y* = *f*(*x*)

*y* = *f*(*x*) shows the relationship

between the input (*x*) and the output (*y*).

If the input is *a*,

the the output is *f*(*a*).

This can be applied to the other inputs:*b*, *c*, *d*, ... .

So, to find the [output],

put the [input] into the *f*(*x*): *f*(input).

## Example 5

*f*(-2) is the output of -2.

So put -2 into *f*(*x*) = 3*x* - 1:*f*(-2) = 3⋅(-2) - 1

= -7.

Let's see what you've solved.

It says

find the output of -2, *f*(-2),

if *f*(*x*) = 3*x* - 1.

After putting -2 into the *f*(*x*),

you found out that *f*(-2) = -7.

## Example 6

*f*(*k*) is the output of *k*.

So put *k* into *f*(*x*) = 2*x* + 5:*f*(*k*) = 2⋅*k* + 5.

And it says *f*(*k*) = 13.

So 2⋅*k* + 5 = 13.

Then *k* = 4.

Linear equations (One variable)

Let's see what you've solved.

It says

find the value of the input *k*

if its output, *f*(*k*), is 13.

After setting *f*(*k*) = 2*k* + 5 = 13,

you found out that *k* = 4.