Section 6.2: Representations of Functions

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.

Property

An equation can define a function by providing a formula to calculate the output value for any given input value. For example, in the equation $h = 1776 - 16t^2$, for any value of the input variable $t$, a unique value of the output variable $h$ can be determined. We say that $h$ is a function of $t$.

Examples

• The equation $P = 2l + 2w$ defines the perimeter of a rectangle as a function of its length and width. However, if width is fixed at 5, $P(l) = 2l + 10$ defines perimeter as a function of length.

• The equation $y = 4x + 3$ defines a linear function. For any $x$-value you choose, you can find a unique corresponding $y$-value.