Lesson 3.5: Relations and Functions

New Concept

A function is a special rule where each input has exactly one output. You will learn to identify functions, find their domain (inputs) and range (outputs), and evaluate them using notation like $f(x)$.

What’s next

Next, you'll work through interactive examples to identify functions, followed by practice problems evaluating them with $f(x)$ notation.

Property

A relation is any set of ordered pairs, $(x, y)$. All the $x$-values in the ordered pairs together make up the domain. All the $y$-values in the ordered pairs together make up the range.
A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.

Examples

• For the relation {(10, A), (20, B), (30, C)}, the domain is {10, 20, 30} and the range is {A, B, C}.

• In the relation {(apple, red), (banana, yellow), (grape, purple), (lime, green)}, the domain is {apple, banana, grape, lime} and the range is {red, yellow, purple, green}.

Property

A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each $x$-value is matched with only one $y$-value.

Examples

• The relation {(-2, 5), (-1, 3), (0, 1), (1, 3), (2, 5)} is a function because every $x$-value is paired with exactly one $y$-value.

• The relation {(1, 5), (1, -5), (4, 10), (4, -10)} is not a function because the $x$-value 1 is paired with both 5 and -5.

Property

For the function $y = f(x)$:

We read $f(x)$ as $f$ of $x$ or the value of $f$ at $x$. The process of finding the value of $f(x)$ for a given value of $x$ is called evaluating the function.

Examples

• For the function $f(x) = 5x - 4$, to evaluate $f(3)$, we substitute 3 for $x$: $f(3) = 5(3) - 4 = 15 - 4 = 11$.

• For the function $g(x) = x^2 + 2x$, to evaluate $g(a)$, we substitute $a$ for $x$: $g(a) = a^2 + 2a$.

Property

For the function $y = f(x)$,
$x$ is the independent variable as it can be any value in the domain
$y$ is the dependent variable as its value depends on $x$

Examples

• The total cost, $C$, of buying $g$ gallons of gas at 3 dollars per gallon is $C(g) = 3g$. The number of gallons $g$ is the independent variable, and the total cost $C$ is the dependent variable.

• The number of hours of daylight, $D$, changes based on the day of the year, $t$. The day $t$ is the independent variable, and the hours of daylight $D$ is the dependent variable.