Learning Objectives
- explain what a function does;
- understand the notation f(x);
- evaluate a function for a given value;
- find an input when the output is known;
- connect function notation to tables and graphs.
Introduction
This lesson develops Functional Notation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🤔 Think About This...
Imagine a machine that doubles any number placed inside it and then adds 3.
A function works like a mathematical machine: it accepts an input, follows a rule and produces an output.
What Does f(x) Mean?
The notation f(x) is read as “f of x”. It means the output of the function f when the input is x.
| Input x | Rule | Output f(x) |
|---|---|---|
| 0 | 2(0) + 3 | 3 |
| 2 | 2(2) + 3 | 7 |
| −1 | 2(−1) + 3 | 1 |
Worked Example 1: Evaluating a Function
Given f(x) = 3x − 5, find f(4).
Replace x with 4.
f(4) = 3(4) − 5.
f(4) = 12 − 5 = 7.
Worked Example 2: Finding the Input
Given f(x) = 2x + 1 and f(x) = 11, find x.
Write 2x + 1 = 11.
Subtract 1: 2x = 10.
Divide by 2: x = 5.
Functions and Graphs
Each input-output pair gives a point on the graph. If f(2) = 7, then the point (2, 7) lies on the graph of y = f(x).
✅ Quick Check
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- For f(x) = x + 4, find f(0), f(3) and f(−2).
- For g(x) = 5x − 1, find g(2).
- Explain what f(6) means.
- Given p(x) = x² + 2, find p(−3).
- Given q(x) = 3x + 4 and q(x) = 19, find x.
- Complete a table for f(x) = 2x − 3 for −2 ≤ x ≤ 3.
- If f(x) = ax + 2 and f(3) = 14, find a.
✅ Check Your Work
Attempt the questions before opening the solutions.
Show answers and working
Foundation
- f(0)=4, f(3)=7, f(−2)=2.
- g(2)=9.
- It means the output of function f when the input is 6.
Developing
- p(−3)=11.
- 3x+4=19, so x=5.
- For x=−2,−1,0,1,2,3, f(x)=−7,−5,−3,−1,1,3.
Challenge
- 3a+2=14, so a=4.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- A function turns an input into an output.
- f(x) names the output of function f.
- Evaluate a function by substituting the input.
- Input-output pairs become coordinates on a graph.