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MathidiaMathematics Academy
Functions and graph transformations
Lesson 31 of 75
Introduction

Function notation

Objective. Evaluate, interpret and build functions using formal notation.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Function notation such as f(x) describes a rule that assigns each input x exactly one output, and f(a) means substitute a wherever x appears in the rule. For f(x) = 3x − 4, f(5) means replace x with 5, giving f(5) = 3(5) − 4 = 11. Function notation also lets us build new functions, such as solving f(x) = 20 by setting 3x − 4 = 20 and solving for x. When evaluating f(x+1) or f(2x), substitute the whole expression carefully in place of x, keeping brackets to avoid errors with negative or multi-term inputs. Always re-check substitutions by writing out the replaced expression in full before simplifying.

Worked example

Given f(x) = 2x² − 3, find f(−2).

Given f(x) = 2x² − 3
Illustrated card showing Given f(x) = 2x² − 3
  1. 1Substitute x = −2 into the rule: f(−2) = 2(−2)² − 3
  2. 2Square first: (−2)² = 4
  3. 3Multiply: 2 × 4 = 8
  4. 4Subtract: 8 − 3 = 5

Answer: 5

Concept mastery

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Practice

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Common mistake. Pupils confuse f(x) with f multiplied by x, writing f(3) = f × 3 instead of substituting 3 into the rule. For f(x) = x² + 1, this wrongly gives 3f instead of f(3) = 9+1=10. Always read f(3) as "the output when x = 3".

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