Functions
Functions form the backbone of GCSE mathematics, yet many Year 10 students struggle when first encountering f(x) = 2x + 3 notation. Understanding how to evaluate functions correctly sets the foundation for advanced topics like transformations and composite functions in Years 11-13.
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Why it matters
Functions model real-world relationships that students encounter daily. A mobile phone contract charging £25 monthly plus £0.15 per text can be expressed as f(x) = 25 + 0.15x, where x represents texts sent. In physics, distance travelled by a falling object follows f(t) = 5t², connecting quadratic functions to motion. Functions help students analyse profit margins in business studies, calculate medication dosages in biology, and understand population growth in geography. The UK National Curriculum emphasises functions from Year 10 through to A-level, where students explore polynomial, rational, and modulus functions. Mastering basic function evaluation in Year 10 prepares students for transformations like f(x+2) and 3f(x) in Year 11, ultimately supporting university-level mathematics and science courses where functions become indispensable analytical tools.
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.
Worked examples
If f(x) = x + 9, find f(7)
Answer: 16
- Substitute x = 7 → f(7) = 7 + 9 = 16 — Replace x with 7 in the expression.
If f(x) = 5x + 4, find f(6)
Answer: 34
- Substitute x = 6 → f(6) = 5 x 6 + 4 = 30 + 4 = 34 — Multiply first, then add or subtract.
If f(x) = x² + 5, find f(6)
Answer: 41
- Calculate x² → 6² = 36 — 6 times 6 equals 36.
- Add 5 → 36 + 5 = 41 — f(6) = 36 + 5 = 41.
Common mistakes
- Students confuse f(3) with 3f, writing f(x) = x + 5 gives f(3) = 3 + 5 = 8 as 3f = 3(x + 5) = 3x + 15 instead.
- Order of operations errors occur when evaluating f(x) = 2x + 3, students calculate f(4) = 2 + 4 + 3 = 9 instead of f(4) = 2(4) + 3 = 11.
- With quadratic functions like f(x) = x² + 2, students write f(3) = 3² + 2 = 6 + 2 = 8 instead of f(3) = 9 + 2 = 11.
- Function notation confusion leads to writing 'f times x equals 2x plus 1' instead of understanding f(x) represents function output.