# 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) = 3x - 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) = 3x - 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) = 2x + 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) = 2k + 5 = 13,
you found out that k = 4.
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