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§ Linear Alg

Functions

§ Linear Alg

Functions

CCSS.8.F3 min read

A function assigns exactly one output value to each input value according to a specific rule. Written as f(x), the function takes an input x and transforms it into an output through operations like multiplication, addition, or squaring. Functions form the foundation of algebra and appear throughout GCSE mathematics, from Year 10 linear graphs to Year 13 polynomial sketching.

§ 01

Why it matters

Functions model countless real-world relationships where one quantity depends on another. Mobile phone bills follow the function C(m) = 25 + 0.15m, where monthly cost depends on minutes used. Population growth follows exponential functions like P(t) = 1000 × 1.03t, predicting city populations over time. Engineers use quadratic functions to design bridges, calculating stress as S(x) = 0.5x² + 10x for beam loading. In physics, distance functions like d(t) = 5t² describe falling objects. Functions appear in GCSE specifications from Year 10 through A-level, connecting algebra to calculus and enabling students to solve complex problems across science, economics, and engineering disciplines.

§ 02

How to solve functions

Functions — Slope & Intercepts

  • A function assigns exactly one output to each input.
  • Slope = (y₂ − y₁) / (x₂ − x₁) for any two points.
  • x-intercept: set y = 0 and solve for x.
  • y-intercept: set x = 0 and solve for y.

Example: Line through (1, 3) and (3, 7): slope = (7−3)/(3−1) = 2.

§ 03

Worked examples

Beginner§ 01

If f(x) = x + 2, find f(8)

Answer: 10

  1. Substitute x = 8 → f(8) = 8 + 2 = 10 — Replace x with 8 in the expression.
Easy§ 02

If f(x) = 5x + 7, find f(4)

Answer: 27

  1. Substitute x = 4 → f(4) = 5 x 4 + 7 = 20 + 7 = 27 — Multiply first, then add or subtract.
Medium§ 03

If f(x) = x² + 2, find f(2)

Answer: 6

  1. Calculate x² → 2² = 4 — 2 times 2 equals 4.
  2. Add 2 → 4 + 2 = 6 — f(2) = 4 + 2 = 6.
§ 04

Common mistakes

  • Confusing function notation: writing f(3) = x + 3 instead of f(3) = 3 + 3 = 6 when evaluating f(x) = x + 3
  • Order of operations errors: calculating f(4) = 2 × 4 + 5 as 2 × 9 = 18 instead of 8 + 5 = 13
  • Function composition mistakes: finding f(g(2)) by calculating f(2) + g(2) = 7 + 5 = 12 instead of first finding g(2) = 5, then f(5) = 11
§ 05

Frequently asked questions

What is the difference between f(x) and y in equations?
f(x) explicitly shows the function relationship and makes substitution clearer, whilst y = mx + c represents the same line but focuses on coordinates. Both describe identical mathematical relationships, but function notation emphasises the input-output process.
How do you find the domain of a function?
The domain contains all possible input values that produce real outputs. For polynomials like f(x) = x², the domain is all real numbers. For fractions like f(x) = 1/x, exclude values making denominators zero (x ≠ 0).
What does function composition f(g(x)) mean?
Function composition applies one function to the output of another. If g(x) = 2x and f(x) = x + 3, then f(g(2)) means first calculate g(2) = 4, then f(4) = 7. Always work from the inside out.
How do you sketch transformed functions like f(x + 2)?
f(x + 2) shifts the original function 2 units left, f(x) + 2 shifts up 2 units, and 2f(x) stretches vertically by factor 2. These transformations appear in Year 11 GCSE, building on basic function graphs.
Can two different inputs give the same output in a function?
Yes, multiple inputs can produce identical outputs. For example, f(x) = x² gives f(2) = 4 and f(-2) = 4. However, each single input must produce exactly one output to qualify as a function.
§ 06

See also

FunctionPolynomialSetDomainMeanFactorMultiple
§ 06

Where to next?

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