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 x1 → y1,
then there's no other x that is paired with y1.
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) = x2 - x - 1,
(g∘f)(x) = ?
Solution (g∘f)(x) = ?
f(x) = 3x
g(x) = x2 - x - 1
(g∘f)(x) = g(f(x))
= g(3x)
= (3x)2 - (3x) - 1
= 9x2 - 3x - 1
g(x) = x2 - x - 1
(g∘f)(x) = g(f(x))
= g(3x)
= (3x)2 - (3x) - 1
= 9x2 - 3x - 1
Close
Example
f(x) = 3x, g(x) = x2 - x - 1,
(f∘g)(x) = ?
Solution (f∘g)(x) = ?
f(x) = 3x
g(x) = x2 - x - 1
(f∘g)(x) = f(g(x))
= f(x2 - x - 1)
= 3(x2 - x - 1)
= 3x2 - 3x - 3
g(x) = x2 - x - 1
(f∘g)(x) = f(g(x))
= f(x2 - x - 1)
= 3(x2 - x - 1)
= 3x2 - 3x - 3
Close
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-1(y)
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-1(5) = ?
Solution f-1(5) = ?
f(2) = 5
→ f-1(5) = 2
→ f-1(5) = 2
Example
f(x) = 2x + 1
f-1(7) = ?
Solution f-1(7) = ?
f-1(7) = x
→ f(x) = 7
2x + 1 = 7 - [1]
2x = 6
x = 3
∴ f-1(7) = 3 - [2]
→ 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-1(7) = x
→ f-1(7) = 3
f-1(7) = x
→ f-1(7) = 3
Close
Example
f(x) = 2x + 4
f-1(x) = ?
Solution f-1(x) = ?
2x + 4 = y
2x = y - 4
x = 12y - 2 - [1]
f-1(y) = 12y - 2 - [2]
f-1(x) = 12x - 2 - [3]
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-1∘f)(x) = x
(f∘f-1)(x) = x
(f∘f-1)(x) = x
Example
f(x) = 4x + 1
(f-1∘f)(3) = ?
Solution (f-1∘f)(3) = ?
(f-1∘f)(3) = 3
Graph: y = f(x) and y = f-1(x)
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