2.3PureCore

Functions

A function is a rule that takes an input and gives exactly one output. Function notation, $f (x)$ , is the language used throughout the rest of A-Level — and for a quadratic it ties neatly to the roots and the graph you already know.

25 min Video by Zeeshan Zamurred Quadratics

Edexcel Quadratics — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Understand function notation f(x) and what it represents
Evaluate a function at a given value
Connect the roots of f(x) to where f(x) = 0
Distinguish a function’s input (domain) from its output (range)

Function notation

Writing $f (x) = x^{2} - 3 x + 2$ names a function $f$ whose rule is “square the input, subtract three lots of it, add two”. The letter in the bracket is just a placeholder for whatever you feed in.

To evaluate, substitute. $f (4)$ means replace every $x$ with $4$ .

f (x) = x^{2} - 3 x + 2 \Rightarrow f (4) = 16 - 12 + 2 = 6

Substitute the input for every

x

to get the output.

1Replace

x

with

- 3

2 (- 3)^{2} + 1

2 \times 9 + 1 = 19

Answer

f (- 3) = 19

Tip — Bracket the input when you substitute: f(−3) uses (−3)², which is +9, not −9.

Roots: where f(x) = 0

The $roots$ of a function are the inputs that make the output zero. For a quadratic these are exactly where the graph crosses the $x$ -axis — so solving $f (x) = 0$ and “finding the roots” are the same task.

f (x) = 0 \Leftrightarrow graph crosses the x -axis

Roots, solutions and

x

-intercepts all describe the same points.

1Set

f (x) = 0

x^{2} - 5 x + 6 = 0

2Factorise:

(x - 2) (x - 3) = 0

Answerroots

x = 2

and

x = 3

Inputs and outputs

The set of allowed inputs is the $domain$ ; the set of resulting outputs is the $range$ . For a basic quadratic $f (x) = x^{2}$ , any real number is a valid input, but the outputs are never negative — so the range is $f (x) \geq 0$ .

Tip — For an upward parabola the range starts at the minimum value (the y-coordinate of the vertex) and goes up.

Formula recap

f (x) = rule applied to the input x

Function notation.

f (a) = substitute x = a

Evaluating a function.

f (x) = 0 \Leftrightarrow roots / x -intercepts

Roots of a function.

Common mistakes to avoid

Reading f(x) as “f times x”.

f(x) means “the function f applied to x”, not multiplication.

Forgetting to bracket a negative input, e.g. f(−2) = 2(−2)² done as −8.

(−2)² = 4, so substitute with brackets to keep signs right.

Key takeaways

f(x) names a function; evaluate it by substituting the input for every x.
The roots are the inputs where f(x) = 0 — the x-axis crossings.
Domain = allowed inputs; range = resulting outputs.

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