Function
How to determine whether a relation is a function and how to find the function value: 5 examples and their solutions.
Example 1
Example
A function is a relation that shows [one x, one y].
Let's see how to show this.
Solution
Draw two sets.
Fill the x values in the left set: 1, 2, 3, 4.
(The left set, x, is called the domain.)
Fill the y values in the right set: 2, 1, 0, 3.
(The right set, y, is called the range.)
Connect each x and the paired y.
1 is paired with 2.
So connect the left 1 and the right 2.
2 is paired with 1.
So connect the left 2 and the right 1.
3 is paired with 0.
So connect the left 3 and the right 0.
4 is paired with 3.
So connect the left 4 and the right 3.
Each left number shows [one x, one y].
So this relation is a function.
Example 2
Example
Solution
Draw two sets.
Fill the x values in the left set: 1, 2, 3, 4.
Fill the y values in the right set: 2, 4, 1.
Connect each x and the paired y.
1 is paired with 2.
So connect the left 1 and the right 2.
2 is also paired with 2.
So connect the left 2 and the right 2.
3 is paired with 4.
So connect the left 3 and the right 4.
4 is paired with 1.
So connect the left 4 and the right 1.
Each left number shows [one x, one y].
So this relation is a function.
Example 3
Example
Solution
Draw two sets.
Fill the x values in the left set: 1, 2, 3, 4.
Fill the y values in the right set: 3, 1, 4, 2.
Connect each x and the paired y.
1 is paired with 3.
So connect the left 1 and the right 3.
2 is paired with 1 and 4.
So connect the left 2 and the right 1.
And connect the left 2 and the right 4.
3 is paired with 4.
So connect the left 3 and the right 4.
4 is paired with 2.
So connect the left 4 and the right 2.
The left 2 shows [one x, two y]:
2 is paired with 1 and 4.
This is not [one x, one y].
So this relation is not a function.
Example 4
Example
f(x) is the y of a function if x = x.
f(x) is read as [f x] or [f of x].
In this example,
if x = x, then y = 3x - 1.
Solution
To find f(-2),
put -2 into f(x) = 3x - 1.
Then f(-2) = 3⋅(-2) - 1.
Then 3⋅(-2) - 1 = -7.
So f(-2) = -7.
This means
if x = -2, then y = -7.
Example 5
Example
Solution
To find f(k),
put k into f(x) = 2x + 5.
Then f(k) = 2k + 5.
It says f(k) = 13.
And f(k) = 2k + 5.
So 2k + 5 = 13.
Solve 2k + 5 = 13.
Then k = 4.
Linear Equation (One Variable)
So k = 4 is the answer.
This means
if x = 4, then y = 13.