Functions: domain, range and composition

A function is a rule that maps each input to exactly one output. In P3 you describe its domain and range and combine functions into composites.

Mappings and functions

One-one: each output comes from exactly one input (for example f(x) = 2x + 1).
Many-one: some outputs come from more than one input (for example f(x) = x², since f(2) = f(-2)).

Both types are functions. A one-many mapping (like x maps to ±sqrt(x)) is not a function.

The domain is the set of inputs; the range is the set of outputs. Describe the range with f(x), not x.

f(x) = x² - 4x + 7 = (x - 2)² + 3 for x ∈ ℝ has range f(x) ≥ 3. A sketch is the safest way to find a range.

fg(x) means f(g(x)): apply g first, then f. Usually fg and gf are different.

The domain of fg is the set of x for which g(x) lies in the domain of f.

f(x) = 2x + 1 and g(x) = x². Find fg(3) and gf(3).

Answer: fg(3) = 19 and gf(3) = 49

Key idea: fg(x) = f(g(x)), so do g first. Give a range in terms of f(x); find it from a sketch or by completing the square.

f(x) = 3x - 2 and g(x) = x² + 1. Evaluate fg(2).
Show answer
13. g(2) = 5, then f(5) = 15 - 2 = 13.
Using f(x) = 3x - 2 and g(x) = x² + 1, find an expression for gf(x), fully expanded.
Show answer
9x² - 12x + 5. gf(x) = g(3x - 2) = (3x - 2)² + 1 = 9x² - 12x + 4 + 1.
f(x) = x² - 4x + 7, x ∈ ℝ. What is the range of f?
- x ≥ 3
- f(x) ≥ 7
- f(x) ≥ 3
- f(x) ≥ 2
Show answer
f(x) ≥ 3. f(x) = (x - 2)² + 3, and the square is never negative, so f(x) ≥ 3.
Which function is one-one for x ∈ ℝ?
- f(x) = x²
- f(x) = 2x³ + 1
- f(x) = cos x
- f(x) = |x|
Show answer
f(x) = 2x³ + 1. 2x³ + 1 is always increasing, so no two inputs share an output.
f(x) = 5 - 2x has domain -1 ≤ x ≤ 3. Find the largest value in the range of f.
Show answer
7. f is decreasing, so the largest value is at x = -1: 5 + 2 = 7.
f(x) = 2x + 1 and g(x) = x² - 3. Solve fg(x) = 13.
Show answer
x = 3 or x = -3. fg(x) = 2(x² - 3) + 1 = 2x² - 5. Then 2x² = 18, so x² = 9.
f(x) = sqrt(x), x ≥ 0, and g(x) = x - 5, x ∈ ℝ. What is the domain of fg?
- x ≥ 0
- x ≥ -5
- x > 5
- x ≥ 5
Show answer
x ≥ 5. fg(x) = sqrt(x - 5), which needs x - 5 ≥ 0.

Last updated 7 November 2026. Spotted a mistake? Tell us.