All skills Math

Learn Index · Functions and Relations Grade 9–10

Domain and range

The domain is the set of inputs; the range is the set of outputs.

Read

Inputs and outputs

  • The domain is the set of all allowed inputs (x-values).
  • The range is the set of all resulting outputs (y-values).

From a list of points

{(1, 4), (2, 7), (3, 4), (5, 9)}

  • Domain: {1, 2, 3, 5}
  • Range: {4, 7, 9} (list repeated values once)

These are discrete sets: separate values only.

From a graph

  • Domain: read left to right. How far does the graph extend horizontally?
  • Range: read bottom to top. How far does it extend vertically?
Graph Domain Range
line y = 2x + 1 all real numbers all real numbers
parabola y = x² all real numbers y ≥ 0
parabola y = −(x − 1)² + 4 all real numbers y ≤ 4
segment from (−2, 1) to (3, 6) −2 ≤ x ≤ 3 1 ≤ y ≤ 6
horizontal line y = 5 all real numbers {5}

From an equation: look for restrictions

Ask: which x-values cause a problem?

Function Restriction Domain
f(x) = 1/x can’t divide by 0 x ≠ 0
f(x) = √x can’t take √ of a negative x ≥ 0
f(x) = √(x − 3) x − 3 ≥ 0 x ≥ 3
f(x) = 5/(x + 2) x + 2 ≠ 0 x ≠ −2

Writing domain and range

Words Set notation Interval notation
all real numbers {x ∈ ℝ} (−∞, ∞)
x is at least 3 {x ∈ ℝ | x ≥ 3} [3, ∞)
x between −2 and 5, inclusive {x ∈ ℝ | −2 ≤ x ≤ 5} [−2, 5]
y is less than 4 {y ∈ ℝ | y < 4} (−∞, 4)

The vertical bar means "such that".

Real-world domains

Context limits the domain.

Example: A car rental costs C(d) = 40d for d days. You can’t rent for negative or fractional days:

  • Domain: {0, 1, 2, 3, …} (whole numbers)
  • Range: {0, 40, 80, 120, …}

Example: A ball’s height h(t) = −5t² + 20t is only meaningful from t = 0 until it lands at t = 4: domain 0 ≤ t ≤ 4, range 0 ≤ h ≤ 20.

Worked examples

Example 1: Domain and range of {(−1, 2), (0, 2), (3, 8)}? D: {−1, 0, 3}; R: {2, 8}.

Example 2: Domain of f(x) = √(2x + 6)? 2x + 6 ≥ 0 → x ≥ −3.

Example 3: Range of y = x² + 5? Smallest value 5 → y ≥ 5.

Common mistakes

  • Swapping domain (x) and range (y).
  • Forgetting restrictions from denominators and square roots.
  • Using continuous intervals for situations that only allow whole numbers.

Try it yourself

  1. Domain and range of {(2, 5), (4, 5), (6, 7)}
  2. Domain of f(x) = 1/(x − 4)
  3. Domain of g(x) = √(x + 1)
  4. Range of y = −x² + 3
  5. Write "x greater than −2" in interval notation.

Answers: 1) D {2, 4, 6}, R {5, 7} 2) x ≠ 4 3) x ≥ −1 4) y ≤ 3 5) (−2, ∞)

Quick recap

  • Domain = all x-values; range = all y-values.
  • Check for restrictions (division by zero, negative square roots).
  • Write answers in set notation or interval notation.