Lesson 4: Identifying Functions and Using Function Notation

New Concept

A function is a mapping between two sets that associates with each element of the first set, the domain, a unique element of the second set, the range.

Why it matters

Functions are the fundamental language of algebra, allowing us to precisely model the relationship between variables. Mastering them is the key to unlocking everything from predicting projectile motion to analyzing economic trends.

What’s next

Next, you’ll learn the vertical line test to visually identify functions and use function notation to work with specific equations.

A function is a mapping between two sets that associates with each element of the first set, the domain, a unique (one and only one) element of the second set, the range. In a function, the y-values may repeat, but the x-values may not.

1. The set {(4, -3), (3, -3)} is a function because each input has only one output. 2. The set {(7, 3), (5, 3), (5, 4)} is not a function because the input 5 has two different outputs (3 and 4). 3. The set {(2, 1), (3, 1), (4, 1)} is a function since each input has exactly one output.

Think of a function as a strict vending machine. Every button you press (the input, or x-value) must give you exactly one, predictable item (the output, or y-value). You can’t press a button for chips and sometimes get a candy bar. However, two different buttons could totally give you the same type of chips!

A graph on the coordinate plane represents the graph of a function provided that any vertical line intersects the graph no more than once.

1. A parabola that opens up, like $y = x^2$, passes the vertical line test and is a function. 2. A circle, like $x^2 + y^2 = 9$, fails the vertical line test and is not a function. 3. A parabola that opens sideways fails the vertical line test and is not a function.

Imagine sliding a pencil held perfectly straight up and down across a graph. If the pencil never touches more than one point on the graphed line at the same time, you've got a function! It’s the ultimate visual check. If your pencil hits two or more spots at once, it fails the test and is just a relation.

Function notation uses parentheses and letters like $f(x)$ or $g(x)$ to distinguish between equations and clearly state which input to use.

1. Given $f(x) = x + 2$ and $g(x) = x - 5$, to find $g(2)$, you use the 'g' equation: $g(2) = 2 - 5 = -3$. 2. Given $p(x) = x^2 - 3x$, to find $p(-3)$, you use the 'p' equation: $p(-3) = (-3)^2 - 3(-3) = 9 + 9 = 18$. 3. If $a(x) = 9 + 6x$, then $a(2)$ is $9 + 6(2) = 21$.

Function notation is like giving your equations cool nicknames to avoid confusion. Instead of saying 'use the first y-equals equation,' you can say 'use $f(x)$'. The value inside the parentheses, like $f(3)$, tells you exactly what number to substitute for $x$. It's a precise and organized way to handle multiple mathematical instructions without mixing them up.

Functions can model real-world scenarios, like costs based on quantity. The cost is the dependent variable (range) because it depends on the number of items you buy, which is the independent variable (domain).

1. A company charges 0.25 dollars per minute. The costs for 2, 3, and 4 minutes are {(2, 0.50), (3, 0.75), (4, 1.00)}, which is a function. 2. Baseball tickets are 25 dollars for the first and 20 dollars for each additional. This creates the function set {(1, 25), (2, 45), (3, 65)}. 3. A buy-one-get-one-free sale on 15.50 dollars t-shirts can be modeled as the function {(1, 15.50), (2, 15.50), (3, 31.00)}.

Think about buying tickets for a baseball game. The total cost depends on how many tickets you buy. If each ticket has a set price, the relationship is a function because two tickets will always have one specific cost, and three tickets will have another. This helps predict expenses for anything from phone bills to concert tickets in a clear way.