M19 · Functions and graph transformations | Edexcel GCSE Maths

Revise the key ideas

f(x) denotes the output of function f for input x; it does not mean f multiplied by x. If f(x) = 3x − 2, then f(4) = 10.
The domain is the permitted input set and the range is the resulting output set. For f(x) = 1x, x = 0 is excluded.
A composite function applies one function then another. fg(x) means f(g(x)): the function nearest x acts first.
Worked example: f(x) = 2x + 1 and g(x) = x². fg(3) = f(9) = 19, but gf(3) = g(7) = 49. Composition order matters.
The inverse f⁻¹ reverses a function; it is not its reciprocal. To find an inverse, write y = f(x), solve for x in terms of y, then exchange the letters.
Worked example: y = 3x − 2 gives x = y + 23, so f⁻¹(x) = x + 23. Applying f and then f⁻¹ restores an allowed input.
A function must be one-to-one on its chosen domain to have an inverse function. x² on all real inputs is not one-to-one; restricting to x ≥ 0 permits inverse √x.

y = f(x) + a moves the graph up a units; y = f(x) − a moves it down a. Every y-coordinate changes by the same amount.
y = f(x − a) moves the graph right a units, whereas y = f(x + a) moves it left a. The sign inside the function often causes mistakes.
A parabola translated two right and three up
Enlarge diagram: A parabola translated two right and three up
Worked example: y = (x − 2)² + 3 is y = x² translated 2 right and 3 up. Its turning point is (2, 3).

y = −f(x) reflects a graph in the x-axis: (x, y) becomes (x, −y). y = f(−x) reflects it in the y-axis: (x, y) becomes (−x, y).
For y = √x, the graph of y = −√x lies below the x-axis with the same non-negative inputs. y = √(−x) instead lies to the left and requires x ≤ 0.
Track a distinctive point and any asymptotes through a transformation. For y = 1x − 2 + 3, asymptotes move from x = 0, y = 0 to x = 2, y = 3.
Check a translated point by substitution. A graph's equation, domain and intercepts must agree with the described movement.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.