Lesson 2: Representations of Functions

Property

An equation defines a function if for any value substituted for the input variable, a unique value for the output variable can be determined.

Examples

• The area of a square is given by the function $A = s^2$. For any side length $s$ you input, you get exactly one area $A$. For $s=4$, the area is always 16.
• A phone plan costs 50 dollars plus 5 dollars per gigabyte of data, $g$. The cost $C$ is a function of data used: $C = 50 + 5g$. For any amount of data, there is only one possible cost.
• The equation $x = y^2$ does not define $y$ as a function of $x$. If $x=9$, $y$ could be 3 or $-3$. Since one input ($x=9$) gives two outputs, it's not a function.

Explanation

An equation acts like a recipe for a function. You provide the input ingredient (the variable, like $x$), and the equation gives you a step-by-step process to calculate exactly one result (the output, $y$).

Property

A function can be described in words, by a table, by a graph, or by an equation.

Function Notation.
Input variable → $f(x) = y$ ← Output variable

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.