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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 shows
one x → one y.
x: domain, input
y: range, output

Example

Function?

xy
12
21
30
43
Solution

Example

Function?

xy
12
22
34
41
Solution

Example

Function?

xy
13
21, 4
34
42
Solution

Vertical Line Test


PassFail
The vertical line test is a way
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

Example

Function?

Solution

Example

Function?

Solution

f(x)

Definition

f(x): a way to write y
by using x.
f(x) is read as
[f of x] or [f x].
To find f(x),
put x
into f(x).

Example

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

Example

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

One-to-one Function

Definition

one x → one unique y
A one-to-one function is a function that shows
one x → one unique y.
So, if x1 → y1,
then there's no other x that is paired with y1.

Horizontal Line Test


PassFail
The horizontal line test is a way
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

Example

One-to-one function?

Solution

Example

One-to-one function?

Solution

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) = x2 - x - 1,
(g∘f)(x) = ?
Solution

Example

f(x) = 3x, g(x) = x2 - x - 1,
(f∘g)(x) = ?
Solution

Example

f(x) = 2x - 1,
(f∘f)(x) = ?
Solution

Example

f(x) = 3x + 1,
(f∘f∘f)(x) = ?
Solution

Inverse Function

Definition

y = f(x)
x = f-1(y)
y = f(x)
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-1(5) = ?
Solution

Example

f(x) = 2x + 1
f-1(7) = ?
Solution

Example

f(x) = 2x + 4
f-1(x) = ?
Solution

Property

(f-1∘f)(x) = x
(f∘f-1)(x) = x

Example

f(x) = 4x + 1
(f-1∘f)(3) = ?
Solution

Graph: y = f(x) and y = f-1(x)

y = f(x), which is x = f-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