Introduction
What Is a Function? is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with what is a function?.
What Is What Is a Function??
What Is a Function? means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding What Is a Function?
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Which graph represents a function?
Why it works: The vertical line test states that a graph is a function if no vertical line intersects it more than once. Options A and D are vertical lines, so they fail. Option C is a circle and fails at multiple points. Option B, a parabola, is a function.
Answer: The parabola (Option B) passes the vertical line test.
Visual Model 2
Question: The table below shows a relation. Is it a function?
| \(x\) | \(y\) |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |
| 4 | 12 |
- A. No, because the outputs are larger than inputs.
- B. Yes, each input has exactly one output.
- C. No, because \(y\) values increase by 3.
- D. Yes, because all values are positive.
Why it works: Each input value (\(1, 2, 3, 4\)) corresponds to exactly one output value (\(3, 6, 9, 12\)). The relationship is \(y = 3x\), making it a function.
Answer: Yes, each input has exactly one output.
Worked Examples
Example 1
Question: Which mapping diagram represents a function?
- Option A has each input mapping to exactly one output.
- Option B has input 1 mapping to two outputs.
- Option C has input 5 mapping to two outputs.
- Option D has input 2 mapping to two outputs.
Answer: Mapping A represents a function.
Example 2
Question: Which table represents a function?
| Table A: | \(x\) | 1 | 2 | 2 | 3 |
|---|---|---|---|---|---|
| \(y\) | 4 | 5 | 6 | 7 |
| Table B: | \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(y\) | 2 | 4 | 6 | 8 |
- A. Table A, because it has more inputs.
- B. Table B, because each input has exactly one output.
- C. Both tables are functions.
- D. Neither table is a function.
- Table A has input 2 appearing twice with different outputs (5 and 6), so it is not a function.
- Table B has each unique input paired with exactly one output.
Answer: Table B is a function.
Example 3
Question: Which graph does NOT represent a function?
- A. A line passing through \((0,2)\) and \((2,6)\).
- B. A horizontal line at \(y = 5\).
- C. A curve shaped like a sideways parabola opening right.
- D. A diagonal line with negative slope.
- A sideways parabola (opening right or left) fails the vertical line test because some vertical lines intersect it twice.
- The other options all pass the vertical line test.
Answer: A sideways parabola fails the vertical line test.
Real-World Word Problems
Problem 1
Question: A math teacher says: "A function is when you input a number and always get the same output." Is this definition correct?
- A. Yes, this is a complete definition.
- B. No, because constant functions like \(y=5\) are not functions.
- C. No, a function only requires that each input has exactly one output, not that different inputs produce the same output.
- D. Yes, as long as the input is positive.
Why it works: The definition of a function only requires that each input has exactly one output. Different inputs can produce different outputs, and some functions are constant while others are not.
Answer: The teacher's definition is incomplete.
Problem 2
Question: A recipe uses 2 cups of flour for every 3 eggs. If \(e\) is the number of eggs, which expression gives the amount of flour needed?
- A. \(f = \frac{2e}{3}\)
- B. \(f = \frac{3e}{2}\)
- C. \(f = 3e + 2\)
- D. \(f = 2e - 3\)
Why it works: The ratio is 2 cups flour : 3 eggs. For \(e\) eggs, flour needed is \(\frac{2}{3} \times e = \frac{2e}{3}\).
Answer: \(f = \frac{2e}{3}\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Which of the following relations is a function?
- A. \(\{(1,2),(1,3),(2,4)\}\)
- B. \(\{(1,2),(2,3),(3,4)\}\)
- C. \(\{(2,5),(2,6),(3,7)\}\)
- D. \(\{(4,1),(4,2),(4,3)\}\)
Question 2
Which statement best defines a function?
- A. A relation where every output has exactly one input.
- B. A relation where every input has exactly one output.
- C. A set of ordered pairs with different \(x\)-coordinates.
- D. Any relation that includes both positive and negative values.
Question 3
Which set of ordered pairs represents a function?
- A. \(\{(0,1),(0,2),(1,3)\}\)
- B. \(\{(2,4),(3,5),(4,6)\}\)
- C. \(\{(1,1),(1,2),(2,2)\}\)
- D. \(\{(5,3),(5,4),(6,5)\}\)
Question 4
What are the domain and range of the relation \(\{(-2,0),(0,4),(2,0),(4,4)\}\)?
- A. Domain: \(\{-2, 2\}\); Range: \(\{0\}\)
- B. Domain: \(\{-2, 0, 2, 4\}\); Range: \(\{0, 4\}\)
- C. Domain: \(\{0, 4\}\); Range: \(\{-2, 0, 2, 4\}\)
- D. Domain: \(\{-2, 0, 2\}\); Range: \(\{4\}\)
Question 5
Which relation is NOT a function?
- A. \(\{(3,1),(3,2),(4,1)\}\)
- B. \(\{(1,5),(2,6),(3,7)\}\)
- C. \(\{(0,0),(1,1),(2,2)\}\)
- D. \(\{(2,8),(3,8),(4,8)\}\)
Question 6
Does the relation \(\{(5,10),(5,15),(6,12)\}\) represent a function?
- A. Yes, because all \(y\)-values are positive.
- B. No, because 5 is paired with two different outputs.
- C. Yes, because there are three ordered pairs.
- D. No, because 10 and 15 are both outputs.
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(\{(1,2),(2,3),(3,4)\}\)
A relation is a function if every input has exactly one output. Option B's inputs \(1, 2, 3\) each appear once. The other options repeat inputs with different outputs.
Question 2
Answer: Every input has exactly one output.
The definition of a function requires that for each input value, there is exactly one corresponding output value. This ensures consistency and predictability.
Question 3
Answer: \(\{(2,4),(3,5),(4,6)\}\)
Each input (\(2, 3, 4\)) maps to exactly one output (\(4, 5, 6\)). The other sets have repeated inputs.
Question 4
Answer: Domain: \(\{-2, 0, 2, 4\}\); Range: \(\{0, 4\}\)
Domain is the set of all input (\(x\)) values: \(-2, 0, 2, 4\). Range is the set of all output (\(y\)) values: \(0, 4\).
Question 5
Answer: \(\{(3,1),(3,2),(4,1)\}\) — input 3 has two outputs.
Option A fails the function test: input 3 maps to both 1 and 2. The other options each have unique input-output pairs.
Question 6
Answer: No, because 5 is paired with two different outputs (10 and 15).
Input 5 maps to both 10 and 15, violating the definition of a function.
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
What Is a Function? becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
