The lesson
This lesson teaches Introduction to Functions. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
One input, one output
A function is like a machine: you put in an input and get exactly one output. If a single input could give two different outputs, it is not a function.
When you study one input, one output, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
The vertical line test
On a graph, if you can draw a vertical line that crosses the graph more than once, the relationship is not a function.
When you study the vertical line test, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
Is the set {(1, 2), (3, 4), (1, 5)} a function?
- 1Look at the inputs: 1 appears twice, with outputs 2 and 5.
- 2Since one input has two different outputs, it is not a function.
Why this matters
Introduction to Functions shows up constantly in a function gives exactly one output for each input. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- One input, one output
- The vertical line test
Video walkthrough
What Is a Function?
Inputs, outputs, and what makes a relationship a function.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfIs the set {(2, 5), (3, 7), (4, 9)} a function?
Step-by-step solution
- 1List inputs: 2, 3, and 4 each appear once.
- 2No input is paired with two different outputs.
- 3Yes, it is a function.
Exercise 2
Try it yourselfIs the set {(1, 4), (1, 6), (2, 8)} a function?
Step-by-step solution
- 1Input 1 appears twice with outputs 4 and 6.
- 2One input has two different outputs.
- 3No, it is not a function.
Exercise 3
Try it yourselfA table shows x: 0, 1, 2 and y: 3, 5, 7. Is this a function?
Step-by-step solution
- 1Each x-value (0, 1, 2) has a single listed y-value.
- 2No x repeats with a different y.
- 3Yes, the table represents a function.
Exercise 4
Try it yourselfUse the vertical line test: a graph is a circle centered at the origin. Is it a function?
Step-by-step solution
- 1A vertical line through x = 0 crosses the circle at (0, r) and (0, −r).
- 2One input (x = 0) would have two outputs.
- 3The circle fails the vertical line test; not a function.
Exercise 5
Try it yourselff(x) = 2x − 1. Find f(4).
Step-by-step solution
- 1Substitute x = 4: f(4) = 2(4) − 1.
- 2Multiply: 8 − 1 = 7.
- 3f(4) = 7.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Introduction to Functions
Question 1 of 4
Which set of ordered pairs is a function?