Lesson 3.1: Functions and Function Notation

New Concept

A function is a predictable relationship where every input has exactly one output. We'll explore how to identify functions from relations, tables, and graphs, and use function notation like $f(x)$ to evaluate and solve real-world problems.

What’s next

You're all set! Next, you'll apply these concepts through a series of practice cards, interactive examples on evaluating functions, and short videos explaining the vertical line test.

Property

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range.
A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range.
To determine if a relationship is a function, identify the input and output values. If each input value leads to only one output value, it's a function. If any input value leads to two or more outputs, it is not a function.

Examples

• The relation {(1, 3), (2, 5), (3, 7)} is a function because each input (1, 2, 3) has exactly one output.
• The relation {(A, 1), (B, 2), (A, 3)} is not a function because the input 'A' is paired with two different outputs, 1 and 3.
• In a school, if each student's name is an input and their assigned homeroom number is the output, this is a function because each student is assigned to only one homeroom.

Explanation

Think of a function like a vending machine. You press one button (the input), and you get exactly one item (the output). If a button gave you two different items, the machine would be broken—just like a relation that isn't a function!

Property

The notation $y = f(x)$ defines a function named $f$. This is read as “$y$ is a function of $x$.” The letter $x$ represents the input value, or independent variable. The letter $y$, or $f(x)$, represents the output value, or dependent variable.

Examples

• If a function $C(g)$ gives the cost of $g$ gallons of gas, $C(10) = 45$ means that 10 gallons of gas cost 45 dollars.
• For the function $f(x) = 4x - 1$, the notation $f(3)$ asks us to replace $x$ with 3. So, $f(3) = 4(3) - 1 = 11$.
• A function $H = g(t)$ gives the height of a rocket in meters $t$ seconds after launch. The statement $g(5) = 150$ means after 5 seconds, the rocket's height is 150 meters.

Explanation

Function notation $f(x)$ is like a recipe's name. The name is 'f', and '(x)' tells you what ingredient to put in. The result, $y$, is the dish you get after following the recipe's instructions to combine the ingredients.

Property

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input.
To evaluate, substitute the input variable in the formula with the value provided and calculate the result.

Examples

• Given $f(x) = x^2 + 5$, to evaluate $f(3)$, we calculate $f(3) = (3)^2 + 5 = 9 + 5 = 14$.
• Given $h(t) = 3t - 2$, to solve for $h(t) = 10$, we set up the equation $3t - 2 = 10$, which gives $3t = 12$, so $t = 4$.
• Given $g(x) = x^2$, to solve for $g(x) = 25$, we set $x^2 = 25$. The solutions are $x = 5$ and $x = -5$, since both inputs produce the output 25.

Explanation

Evaluating is like asking, "If I drive for 2 hours at 60 mph, how far do I go?" Solving is the reverse: "If I drove 120 miles at 60 mph, how long did it take?" Evaluating gives one answer; solving might give several.

Property

A one-to-one function is a function in which each output value corresponds to exactly one input value.

Examples

• The function $f(x) = x - 7$ is one-to-one. For any output, like 10, there is only one input that works: $x=17$.
• The function $g(x) = x^2$ is not one-to-one. The output 16 corresponds to two different inputs, $x = 4$ and $x = -4$.
• A list pairing each person with their unique social security number is a one-to-one function. Each person has one number, and each number belongs to only one person.

Explanation

A one-to-one function is a perfect pairing. Not only does every input have a unique output, but every output comes from only one unique input. No sharing allowed, either way! It's like a reserved parking spot for every car.

Property

The vertical line test can be used to determine whether a graph represents a function.
If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function.
The horizontal line test is used to determine if the graph of a function is one-to-one.
If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Examples

• A circle on a graph fails the vertical line test because a vertical line can cross it twice. Therefore, a circle's graph does not represent a function.
• The graph of $y = |x|$ (a V-shape) passes the vertical line test, so it is a function. However, it fails the horizontal line test, so it is not one-to-one.
• The graph of a straight line like $y = 2x + 1$ passes both the vertical and horizontal line tests. Therefore, it is a one-to-one function.

Explanation

The vertical line test checks if any input has multiple outputs (not a function). The horizontal line test checks if any output comes from multiple inputs (not one-to-one). Think of it as scanning the graph with a ruler!