1.2 Functions

Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 1: Pure Mathematics 1 · official syllabus

FUNCTIONS LAB

Learn it by playing it

Four short missions on one live function machine. When a mission glows green, read why it worked, then take the next one.

1234
Mission 1 of 4

Break the domain

Every function in this lab is a machine that eats x and spits out y - but this one refuses exactly one input. Slide c and put the graph's vertical gap exactly at x = 2.

vertical gap at x = 0.00 - target 2 (domain: all reals except 0.00)

Study notes

Played the lab above? These notes are the reference version of what you just felt - use them to check exact formulas and the exam wording.

Functions, domain and range 1.2

  • A function is a rule that gives exactly one output for each allowed input. The domain is the set of allowed inputs; the range is the set of outputs the function actually produces.
  • State the domain whenever it is restricted, e.g. f : x -> 1/(x - 2) for x not equal to 2.
  • Find simple ranges from a sketch or a known shape: g : x -> x² + 1 for x in the reals has range g(x) >= 1, because x² >= 0.
  • For a quadratic, complete the square to read off the range: f(x) = 2x² - 8x + 3 = 2(x - 2)² - 5, so the least value is -5 and the range is f(x) >= -5.

Composite functions 1.2

  • fg(x) means f(g(x)): apply g FIRST, then f. fg and gf are usually different.
  • The composite gf can only be formed when the range of f lies within the domain of g - every output of f must be an allowed input of g.
  • Work composites numerically from the inside out: with f(x) = 3x - 2 and g(x) = x + 4, gf(2) = g(4) = 8.

One-one functions and inverses 1.2

  • A function is one-one when each output comes from exactly one input. x² on all reals is not one-one (2 and -2 both give 4); restricting the domain to x >= 0 makes it one-one.
  • Only a one-one function has an inverse. To find f⁻¹: write y = f(x), make x the subject, then swap x and y. For f(x) = 3x - 5, f⁻¹(x) = (x + 5) / 3.
  • When the domain was restricted, choose the root that matches it: for h(x) = (2x + 3)² - 4 with x >= -3/2, 2x + 3 >= 0, so take the positive square root and h⁻¹(x) = (sqrt(x + 4) - 3) / 2.
  • The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
  • ff⁻¹(x) = x and f⁻¹f(x) = x wherever the composites exist.

Graphs of a function and its inverse 1.2

  • The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x. Mark the mirror line on sketches.
  • Where the graph of f crosses y = x, it also meets the graph of f⁻¹.

Transformations of graphs 1.2

  • y = f(x) + a is a translation of a units parallel to the y-axis (up when a > 0).
  • y = f(x + a) is a translation of -a units parallel to the x-axis - the shift is opposite to the sign inside the bracket.
  • y = af(x) is a stretch of scale factor a parallel to the y-axis; y = f(ax) is a stretch of scale factor 1/a parallel to the x-axis.
  • y = -f(x) reflects in the x-axis; y = f(-x) reflects in the y-axis. Use the words translation, reflection and stretch in descriptions.
  • Combine transformations one at a time in order: y = x² -> y = 2(x - 3)² + 1 is a translation 3 units right, then a stretch of scale factor 2 parallel to the y-axis, then a translation 1 unit up.

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