A function f maps each element of its domain to exactly one output value, written f: x ↦ f(x) or y = f(x). These notes cover function notation, domain and range, composite and inverse functions, and the key graphs: the rectangular hyperbola, the modulus function and graph transformations.
Functions underpin almost every A-Math topic that follows. Differentiation, integration and logarithms all rely on fluency with notation, composition and inverses.
What these notes cover
- •What is a function
- •Domain, range and codomain
- •Vertical line test
- •Composite functions
- •Inverse functions
- •Horizontal line test
- •Rectangular hyperbola
- •Modulus function
- •Graph transformations
- •Common mistakes
- •Summary
Key definitions
- Function
- A rule f that maps each element of its domain to exactly one output value, written f: x ↦ f(x) or y = f(x). A relation is a function if and only if every vertical line meets its graph at most once (vertical line test).
- Domain
- The set of permitted input values, either given explicitly or taken as the largest set for which f(x) is defined.
- Range
- The set of all output values actually produced by f.
- Codomain
- The declared target set of a function. It differs from the range, which is the set of values f actually outputs.
- One-to-one (injective) function
- A function in which each output comes from exactly one input. It is checked with the horizontal line test: every horizontal line must cut the graph at most once.
Key formulas
Composite function
fg(x) = f(g(x))
Apply g first, then f. Order matters, so fg ≠ gf in general. fg exists if and only if Range(g) ⊆ Domain(f).
Inverse function property
f⁻¹ exists only if f is one-to-one. The domain of f⁻¹ equals the range of f, and the graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
Rectangular hyperbola
Two branches lying in opposite quadrants; the x- and y-axes are asymptotes.
Translated rectangular hyperbola
Shifts the asymptotes to x = h and y = k.
Modulus function
y = |f(x)|
Any portion of y = f(x) below the x-axis is reflected upward, so the range is always ≥ 0. y = |x − a| is a V-shape with vertex at (a, 0).
Horizontal scaling
y = f(ax)
Horizontal scale by factor 1 ÷ a; the graph is compressed toward the y-axis if a > 1.
Worked example
Find f⁻¹(x) for f(x) = (3x − 1) ÷ (x + 2), x ≠ −2.
- 1Let y = (3x − 1) ÷ (x + 2).
- 2Multiply both sides by (x + 2): y(x + 2) = 3x − 1, so xy + 2y = 3x − 1.
- 3Collect the x terms on one side: xy − 3x = −2y − 1.
- 4Factorise and divide: x(y − 3) = −2y − 1, so x = (−2y − 1) ÷ (y − 3).
- 5Replace y with x: f⁻¹(x) = (−2x − 1) ÷ (x − 3).
- 6Since f(x) = 3 has no solution, 3 is excluded from the range of f, so the domain of f⁻¹ is x ∈ ℝ, x ≠ 3.
Watch out: The domain of f⁻¹ is the range of f, not the domain of f. Here it excludes 3, not −2.
Common mistakes
Confusing fg and gf
Wrong: Writing gf when fg is meant, so f is applied first.
Right: Read the composite from right to left: fg means apply g first, then f.
Wrong domain for f⁻¹
Wrong: Giving the domain of f⁻¹ as the domain of f.
Right: The domain of f⁻¹ equals the range of f.
Horizontal shift direction
Wrong: Shifting y = f(x + 3) right by 3.
Right: y = f(x + a) moves the graph left and y = f(x − a) moves it right. Ask what value of x makes the bracket zero; the shift goes opposite to the sign inside the bracket.
Questions students ask
When does a composite function fg exist?
fg exists if and only if Range(g) ⊆ Domain(f). Find the range of g first, then check that every one of those values is a permitted input for f. Remember that fg(x) = f(g(x)) means apply g first, then f.
How do you find the inverse of a function?
First check that f is one-to-one using the horizontal line test; if it fails, restrict the domain. Then write y = f(x), make x the subject and replace y with x to get f⁻¹(x). Finally, state the domain of f⁻¹, which equals the range of f.
What is the difference between range and codomain?
The range is the set of values f actually outputs, while the codomain is the declared target set. In exam questions, find the range by sketching the graph or by solving for x in terms of y.
How do you sketch a modulus graph y = |f(x)|?
Sketch y = f(x) first, then reflect any portion lying below the x-axis upward, never downward. The range of y = |f(x)| is always ≥ 0, and y = |x − a| is a V-shape with vertex at (a, 0).
In the full PDF
- •Figure 1: sketch of the rectangular hyperbola y = 1 ÷ x showing both branches and asymptotes
- •Worked example on computing fg(x) and finding its range
- •Worked example on sketching y = |2x − 4| and stating its range
- •Full graph transformations table covering translations, reflections and stretches
- •Remaining common mistakes, including the one-to-one check and modulus fold direction
- •Five-point summary of the whole topic