# Function (Math)

See how to solve function (math) problems.

20 examples and their solutions.

## Function

### Definition

one x → one y

A function is a relation that showsone x → one y.

x: domain, input

y: range, output

### Example

Function?

Solution x | y |
---|---|

1 | 2 |

2 | 1 |

3 | 0 |

4 | 3 |

function

Draw the domain X:

1, 2, 3, 4.

Draw the range Y:

2, 1, 0, 3.

Connect each paired x and y.

This figure shows

one x → one y.

So this relation is a function.

1, 2, 3, 4.

Draw the range Y:

2, 1, 0, 3.

Connect each paired x and y.

This figure shows

one x → one y.

So this relation is a function.

Close

### Example

Function?

Solution x | y |
---|---|

1 | 2 |

2 | 2 |

3 | 4 |

4 | 1 |

function

This figure shows

one x → one y.

So this relation is a function.

one x → one y.

So this relation is a function.

Close

### Example

Function?

Solution x | y |
---|---|

1 | 3 |

2 | 1, 4 |

3 | 4 |

4 | 2 |

not a function

x = 2 → y = 1, 4

This shows

one x → two y.

So this relation is not a function.

This shows

one x → two y.

So this relation is not a function.

Close

### Vertical Line Test

PassFail

to see if a graph is a function.

Think of a vertical line (= one x).

Move the vertical line

from the left to the right.

If the vertical line and the graph

intersect at one point (= one (x, y)),

then the graph passes the test.

(= The graph is a function.)

If the vertical line and the graph

intersect at more than one point (≠ one (x, y)),

then the graph fails the test.

(= The graph is not a function.)

### Example

Function?

Solution The vertical line and the graph

intersect at one point.

So the graph passes the vertical line test.

So the graph is a function.

intersect at one point.

So the graph passes the vertical line test.

So the graph is a function.

Close

### Example

Function?

Solution If the vertical line is on the left endpoint,

the vertical line and the graph

intersect at one point.

the vertical line and the graph

intersect at one point.

↓

But, as the vertical line moves to the right,

the vertical line and the graph

intersect at two points.

So the graph fails the vertical line test.

So the graph is not a function.

the vertical line and the graph

intersect at two points.

So the graph fails the vertical line test.

So the graph is not a function.

Close

### Example

Function?

Solution The vertical line and the graph

intersect at one point.

So the graph passes the vertical line test.

So the graph is a function.

intersect at one point.

So the graph passes the vertical line test.

So the graph is a function.

Close

## f(x)

### Definition

f(x): a way to write y

by using x.

f(x) is read asby using x.

[f of x] or [f x].

To find f(x),

put x

into f(x).

### Example

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

Solution f(2) = 3⋅2 - 1 - [1]

= 6 - 1

= 5

= 6 - 1

= 5

[1]

Put 2 into f(x).

Close

### Example

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

Solution f(k) = 2⋅k + 5 = 13 - [1] [2] [3]

2k = 8

k = 4

2k = 8

k = 4

[1]

Put k into f(x).

→ f(k) = 2⋅k + 5

→ f(k) = 2⋅k + 5

[2]

f(k) = 2k + 5

f(k) = 13→ 2k + 5 = 13

f(k) = 13→ 2k + 5 = 13

[3]

Solve 2k + 5 = 13.

Linear Equation (One Variable)

Linear Equation (One Variable)

Close

## One-to-one Function

### Definition

one x → one unique y

A one-to-one function is a function that showsone x → one unique y.

So, if x

_{1}→ y

_{1},

then there's no other x that is paired with y

_{1}.

### Horizontal Line Test

PassFail

to see if a graph is a one-to-one function.

Think of a horizontal line (= one unique y).

Move the horizontal line

from the top to the bottom.

If the horizontal line and the graph

intersect at one point (= one (x, y)),

then the graph passes the test.

(= The graph is a one-to-one function.)

If the horizontal line and the graph

intersect at more than one point (≠ one (x, y)),

then the graph fails the test.

(= The graph is not a one-to-one function.)

### Example

One-to-one function?

Solution First, check if the graph is a function

by doing the vertical line test.

(if the graph is not a function,

it cannot be a one-to-one 'function'.)

The graph passes the vertical line test.

So the graph is a function.

by doing the vertical line test.

(if the graph is not a function,

it cannot be a one-to-one 'function'.)

The graph passes the vertical line test.

So the graph is a function.

↓

Then see if the graph

passes the horizontal line test.

The horizontal line and the graph

intersect at one point.

So the graph passes the horizontal line test.

So the graph is a one-to-one function.

passes the horizontal line test.

The horizontal line and the graph

intersect at one point.

So the graph passes the horizontal line test.

So the graph is a one-to-one function.

Close

### Example

One-to-one function?

Solution The graph passes the vertical line test.

So the graph is a function.

So the graph is a function.

↓

If the horizontal line is on the bottom endpoint,

the vertical line and the graph

intersect at one point.

the vertical line and the graph

intersect at one point.

↓

But, as the horizontal line moves upward,

the horizontal line and the graph

intersect at two points.

So the graph fails the horizontal line test.

So the graph is not a one-to-one function.

the horizontal line and the graph

intersect at two points.

So the graph fails the horizontal line test.

So the graph is not a one-to-one function.

Close

### Example

One-to-one function?

Solution The graph passes the vertical line test.

So the graph is a function.

So the graph is a function.

↓

If the horizontal line is like this,

the vertical line and the graph

intersect at one point.

the vertical line and the graph

intersect at one point.

↓

But, as the horizontal line moves upward,

the horizontal line and the graph

intersect at two points.

So the graph fails the horizontal line test.

So the graph is not a one-to-one function.

the horizontal line and the graph

intersect at two points.

So the graph fails the horizontal line test.

So the graph is not a one-to-one function.

Close

## Composite Function

### Formula

(g∘f)(x) = g(f(x))

To find g(f(x)),put f(x)

into g( ).

### Example

f(x) = 3x, g(x) = x

(g∘f)(x) = ?

Solution ^{2}- x - 1,(g∘f)(x) = ?

f(x) = 3x

g(x) = x

(g∘f)(x) = g(f(x))

= g(3x)

= (3x)

= 9x

g(x) = x

^{2}- x - 1(g∘f)(x) = g(f(x))

= g(3x)

= (3x)

^{2}- (3x) - 1= 9x

^{2}- 3x - 1Close

### Example

f(x) = 3x, g(x) = x

(f∘g)(x) = ?

Solution ^{2}- x - 1,(f∘g)(x) = ?

f(x) = 3x

g(x) = x

(f∘g)(x) = f(g(x))

= f(x

= 3(x

= 3x

g(x) = x

^{2}- x - 1(f∘g)(x) = f(g(x))

= f(x

^{2}- x - 1)= 3(x

^{2}- x - 1)= 3x

^{2}- 3x - 3Close

### Example

f(x) = 2x - 1,

(f∘f)(x) = ?

Solution (f∘f)(x) = ?

(f∘f)(x) = f(f(x))

= f(2x - 1)

= 2(2x - 1) - 1

= 4x - 2 - 1

= 4x - 3

= f(2x - 1)

= 2(2x - 1) - 1

= 4x - 2 - 1

= 4x - 3

Close

### Example

f(x) = 3x + 1,

(f∘f∘f)(x) = ?

Solution (f∘f∘f)(x) = ?

(f∘f∘f)(x) = f(f(f(x)))

= f(f(3x + 1))

= f(3(3x + 1) + 1)

= f(9x + 3 + 1)

= f(9x + 4)

= 3(9x + 4) + 1

= 27x + 12 + 1

= 27x + 13

= f(f(3x + 1))

= f(3(3x + 1) + 1)

= f(9x + 3 + 1)

= f(9x + 4)

= 3(9x + 4) + 1

= 27x + 12 + 1

= 27x + 13

Close

## Inverse Function

### Definition

y = f(x)

→ x = f

y = f(x)→ x = f

^{-1}(y)If you change this to [x = ...],

then you get x = f

^{-1}(y).

This f

^{-1}is the inverse function of f.

f

^{-1}is a way to write x

by using y.

f

^{-1}(x) is read as

[f inverse x].

### Example

f(2) = 5

f

Solution f

^{-1}(5) = ? f(2) = 5

→ f

→ f

^{-1}(5) = 2Close

### Example

f(x) = 2x + 1

f

Solution f

^{-1}(7) = ? f

→ f(x) = 7

2x + 1 = 7 - [1]

2x = 6

x = 3

∴ f

^{-1}(7) = x→ f(x) = 7

2x + 1 = 7 - [1]

2x = 6

x = 3

∴ f

^{-1}(7) = 3 - [2][1]

f(x) = 7

f(x) = 2x + 1

→ 2x + 1 = 7

f(x) = 2x + 1

→ 2x + 1 = 7

[2]

x = 3

f

→ f

f

^{-1}(7) = x→ f

^{-1}(7) = 3Close

### Example

f(x) = 2x + 4

f

Solution f

^{-1}(x) = ? 2x + 4 = y

2x = y - 4

x = 12y - 2 - [1]

f

f

2x = y - 4

x = 12y - 2 - [1]

f

^{-1}(y) = 12y - 2 - [2]f

^{-1}(x) = 12x - 2 - [3][1]

Change y = 2x + 4 to [x = ...].

[2]

x → f

^{-1}(y)[3]

Switch x and y.

Close

### Property

(f

(f∘f

^{-1}∘f)(x) = x(f∘f

^{-1})(x) = x### Example

f(x) = 4x + 1

(f

Solution (f

^{-1}∘f)(3) = ? (f

^{-1}∘f)(3) = 3Close

### Graph: y = f(x) and y = f^{-1}(x)

^{-1}(y),

and its inverse function y = f

^{-1}(x)

are symmetric about y = x.

(because x and y are switched)

So,

y = f(x) passes the horizontal line test.

→ y = f

^{-1}(x) passes the vertical line test.

→ y = f

^{-1}(x) exists.

### Example

Determine whether the given function has an inverse function.

Solution The given graph passes the horizontal line test.

So the given graph has an inverse function.

So the given graph has an inverse function.

Close