Lesson 1: Functions

New Concept

A function is a special relationship where each input value determines exactly one output value. We will explore how to identify and interpret functions represented by tables, graphs, and equations using notation like $f(x)$.

What’s next

Now, you'll work through interactive examples to identify functions. Then, you'll use practice cards to master evaluating them from tables, graphs, and equations.

Property

A function is a relationship between two variables for which a exactly one value of the output variable is determined by each value of the input variable.
The variable used for the input is the input variable, and the variable for the output is the output variable.

Examples

• The area $A$ of a circle is a function of its radius $r$, given by the formula $A = \pi r^2$. For any radius you choose, there is only one possible area.

• The total cost of buying concert tickets is a function of the number of tickets purchased. If each ticket costs 50 dollars, the cost $C$ for $n$ tickets is $C = 50n$.

Property

When we use a table to describe a function, the first variable in the table (the left column or top row) is the input variable, and the second variable is the output. We say that the output variable is a function of the input. A table does not represent a function if a single input value corresponds to more than one output value.

Examples

• A table shows hours worked ($h$) and earnings ($E$). If for every value of $h$ there is only one value for $E$, then $E$ is a function of $h$. For example, if $h=5$ always results in $E=75$ dollars.

• This table defines $y$ as a function of $x$ because each $x$-value has exactly one corresponding $y$-value: $|x: 1, 2, 3|, |y: 5, 10, 15|$.

Property

We can also use a graph to define a function. The input variable is displayed on the horizontal axis, and the output variable on the vertical axis. For a graph to represent a function, every vertical line drawn on the graph can intersect the curve at most one time. This is known as the vertical line test.

Examples

• A graph showing the temperature in a city over a 24-hour period represents a function, because at any specific time (input), there is only one temperature (output).

• The graph of a parabola opening upwards or downwards, like $y = x^2$, is a function because any vertical line will only cross the graph once.

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.

Property

We use a letter like $f$ or $g$ to name a function. The notation $f(x)$, read '$f$ of $x$', represents the output value of the function $f$ when the input is $x$. If $y$ is the output variable, we can write $y = f(x)$. The parentheses in $f(x)$ do not indicate multiplication.

Function Notation:
Input variable
↓
$f(x) = y$
↑
Output variable

Examples

• Instead of writing 'the area $A$ for a radius $r$ is $\pi r^2$', we can write $A(r) = \pi r^2$. The notation $A(3)$ asks for the area of a circle with a radius of 3.

Property

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function. To evaluate a function described by an equation, we substitute the given input value into the equation to find the corresponding output, or function value.

Examples

• Given the function $f(x) = 5x - 2$, to evaluate $f(3)$, we substitute $x=3$ to get $f(3) = 5(3) - 2 = 15 - 2 = 13$.

• For the function $g(t) = t^2 + 10$, evaluating at $t=-4$ means calculating $g(-4) = (-4)^2 + 10 = 16 + 10 = 26$.