Functions — Domain, Range, and Composition
What a function is, how to characterise it by its domain and range, and how functions combine through composition.
What a Function Is
A function is a rule that assigns to each input exactly one output. One input → one output. That’s the whole definition.
f(x) = 2x + 1
For any x you put in, you get exactly one value out. f(3) = 7. f(−1) = −1. No input produces two different outputs.
Notation: f(x) is read “f of x.” The letter f names the function; x is the input. The output is f(x).
The vertical line test
A graph represents a function if and only if every vertical line crosses it at most once. If a vertical line hits the graph at two points, two outputs correspond to one input — not a function.
Domain
The domain is the set of all valid inputs — all values x can take.
Three things that restrict a domain:
- Division by zero — the denominator cannot be zero
- Even roots of negatives — √(negative) is not real
- Logarithms of non-positives — log(x) requires x > 0
f(x) = 1/(x − 3) domain: x ≠ 3, i.e., (−∞,3) ∪ (3,∞)
f(x) = √(x − 2) domain: x ≥ 2, i.e., [2, ∞)
f(x) = log(x + 5) domain: x > −5, i.e., (−5, ∞)
f(x) = x² + 1 domain: all reals, (−∞, ∞)
If no domain is stated, assume all real numbers that make the expression defined.
Range
The range is the set of all possible outputs — the values f(x) can actually take.
Finding the range is often harder than finding the domain. Strategies:
- Think about what values the function can and cannot produce
- Solve x = f(y) for y, then find the domain of that expression
- For simple functions, read the range from a graph
f(x) = x² range: [0, ∞) — squares are never negative
f(x) = x² − 4 range: [−4, ∞) — shifts the parabola down by 4
f(x) = √x range: [0, ∞)
f(x) = 1/x range: all reals except 0, (−∞,0) ∪ (0,∞)
f(x) = sin(x) range: [−1, 1]
Types of Functions
Linear: f(x) = mx + b — straight line, constant rate of change. m is the slope, b is the y-intercept.
Quadratic: f(x) = ax² + bx + c — parabola. Opens up if a > 0, down if a < 0.
Polynomial: f(x) = aₙxⁿ + … + a₁x + a₀ — sums of power terms. Domain always all reals.
Rational: ratio of polynomials. Denominator restrictions apply.
Exponential: f(x) = aˣ — rapid growth or decay. Domain all reals, range (0, ∞) if a > 0.
Logarithmic: f(x) = log(x) — inverse of exponential. Domain (0, ∞), range all reals.
Trigonometric: sin, cos, tan — periodic functions. Covered in the trigonometry note.
Transformations
Starting from a base function f(x), transformations shift, stretch, or reflect it:
| Transformation | Effect |
|---|---|
| f(x) + k | shift up k units |
| f(x) − k | shift down k units |
| f(x + h) | shift left h units |
| f(x − h) | shift right h units |
| −f(x) | reflect over x-axis |
| f(−x) | reflect over y-axis |
| a·f(x) | stretch vertically by a |
| f(ax) | compress horizontally by a |
g(x) = (x − 3)² + 2
This is f(x) = x² shifted right 3, up 2. Vertex at (3, 2).
Transformations compound — apply inside the function first (horizontal effects), then outside (vertical effects).
Inverse Functions
The inverse function f⁻¹ undoes what f does. If f(a) = b, then f⁻¹(b) = a.
f(x) = 2x + 3
y = 2x + 3
x = (y − 3)/2 (solve for x)
f⁻¹(x) = (x − 3)/2
Check: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Graphically: the inverse is the reflection of f over the line y = x.
Not every function has an inverse. f(x) = x² doesn’t — f(2) = f(−2) = 4, so you can’t reverse from 4 to a unique input. A function must be one-to-one (each output comes from exactly one input) to have an inverse. The horizontal line test checks this on a graph.
Composition
Function composition feeds the output of one function into another:
(f ∘ g)(x) = f(g(x))
First apply g, then apply f to the result.
f(x) = x + 1
g(x) = x²
(f ∘ g)(x) = f(g(x)) = f(x²) = x² + 1
(g ∘ f)(x) = g(f(x)) = g(x + 1) = (x + 1)²
Order matters — f ∘ g ≠ g ∘ f in general.
Domain of a composition
The domain of f ∘ g is all x in the domain of g such that g(x) is in the domain of f.
f(x) = √x (domain: x ≥ 0)
g(x) = x − 4
(f ∘ g)(x) = √(x − 4)
domain: x − 4 ≥ 0 → x ≥ 4
Decomposing functions
Recognising a function as a composition is useful in calculus (chain rule) and when designing systems:
h(x) = (3x + 1)⁵
= f(g(x)) where g(x) = 3x + 1 and f(x) = x⁵
Even and Odd Functions
Even function: f(−x) = f(x) — symmetric about the y-axis. Examples: x², cos(x), |x|.
Odd function: f(−x) = −f(x) — symmetric about the origin. Examples: x³, sin(x), x.
Most functions are neither. These properties matter in integration (odd functions integrate to zero over symmetric intervals) and signal processing (even/odd decomposition).