Revision notes, worked examples and methods for functions and graph transformations.
Notes and quizzes ready · 50 questions · Video coming soon.
This topic is Higher-only.
Revise the key ideas
Higher — function notation
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.
Higher — graph translations
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
Worked example: y = (x − 2)2 + 3 is y = x2 translated 2 right and 3 up. Its turning point is (2, 3).
Higher — graph reflections
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.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
Mr Kidman's Classroom