The lesson
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.
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.
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
A function gives exactly one output for each input.
These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.
Common mistakes to avoid
Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.
Re-reading the question too quickly. Underline what you are solving for before you start calculating.
Key ideas from this lesson
- One input, one output
- The vertical line test
Video walkthroughs
What Is a Function?
Inputs, outputs, and what makes a relationship a function.
Watch on YouTube
Evaluating Functions from Graphs
Read inputs and outputs from a function graph.
Watch on YouTube
Graphs of Linear Equations
See how ordered pairs define a function on the coordinate plane.
Watch on YouTube
Dependent & Independent Variables
Identify input and output in function relationships.
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?