Practice Maths

L41 — Domain, Range and Types of Functions

Key Terms

Domain
The set of all valid input values (x); the natural domain excludes values where the function is undefined.
Range
The set of all possible output values (y) produced by the function for the given domain.
Natural domain
The largest set of x values for which the function is defined; exclude values causing division by zero or negative radicands.
Piecewise function
A function defined by different rules on different intervals of the domain.
One-to-one function
Each output is produced by at most one input — passes the horizontal line test; required for an inverse to exist.
Many-to-one function
Some outputs are produced by more than one input (e.g. f(x) = x²); does not have an inverse on its full domain.

Domain and range

TermMeaningHow to find it
DomainAll valid input (x) valuesAsk: what can x be without breaking the function?
RangeAll resulting output (y) valuesAsk: what y-values does the function actually produce?
Natural domainLargest possible domain unless restrictedExclude: denominator = 0, negative under √

Common function types and their domains/ranges

Function typeExampleNatural domainRange
Linearf(x) = 2x + 1All reals (−∞, ∞)All reals
Quadraticf(x) = x² − 3All reals[−3, ∞)
Square rootf(x) = √xx ≥ 0 → [0, ∞)[0, ∞)
Rationalf(x) = 1/xx ≠ 0 → (−∞,0) ∪ (0,∞)y ≠ 0
Absolute valuef(x) = |x|All reals[0, ∞)
Exponentialf(x) = 2xAll reals(0, ∞)
Linear: y = x Quadratic: y = x² Square root: y = √x Absolute value: y = |x| Four common function shapes
Common function types and their characteristic shapes.
Hot Tip: To state the natural domain, ask: what values of x would cause problems? Division by zero → exclude that value. Square root of a negative → require the expression under the root ≥ 0.

Worked Example 1 — Finding the natural domain

Find the natural domain of: (a) f(x) = √(2x − 6); (b) g(x) = 3/(x − 4); (c) h(x) = √(x + 1) / (x − 3).

(a) Need 2x − 6 ≥ 0 ⇒ x ≥ 3. Domain: x ≥ 3 or [3, ∞).

(b) Need x − 4 ≠ 0 ⇒ x ≠ 4. Domain: all reals except x = 4.

(c) Need x + 1 ≥ 0 (so x ≥ −1) AND x ≠ 3. Domain: x ≥ −1, x ≠ 3.

Worked Example 2 — Finding the range

Find the range of: (a) f(x) = (x − 2)² + 5; (b) g(x) = √(x + 3).

(a) (x−2)² ≥ 0 always, so f(x) = (x−2)² + 5 ≥ 5. Range: y ≥ 5 or [5, ∞). Minimum value 5 occurs at x = 2.

(b) √(x+3) ≥ 0 always. As x increases from −3, output increases from 0. Range: y ≥ 0 or [0, ∞).

Worked Example 3 — Reading domain and range from a graph

A graph shows a parabola with vertex at (1, −2), x-intercepts at x = −1 and x = 3, drawn for −3 ≤ x ≤ 5. State the domain and range.

Domain (x-values shown): −3 ≤ x ≤ 5.

Range: vertex gives minimum y = −2. At endpoints, x = −3 gives y = (−3−1)² − 2 = 14, x = 5 gives y = (5−1)² − 2 = 14. Maximum y = 14. Range: −2 ≤ y ≤ 14.

See Answers ➔

  1. Natural domain. Fluency

    State the natural domain of each function.

    • (a) f(x) = 4x − 7
    • (b) g(x) = √(x − 5)
    • (c) h(x) = 1/(x + 3)
    • (d) k(x) = √(9 − x²)

  2. Range of functions. Fluency

    • (a) f(x) = x² − 4. State the range.
    • (b) g(x) = −x² + 1. State the range.
    • (c) h(x) = √x. State the range.
    • (d) f(x) = 3x + 2. State the range.

  3. Function types. Fluency

    • (a) Classify each function type: f(x) = 2x − 3; g(x) = x² + 1; h(x) = 3x; k(x) = √(x + 2).
    • (b) Which of the functions in (a) have a restricted natural domain?
    • (c) Which of the functions in (a) have a restricted range?
    • (d) The function f(x) = |2x − 4| — state the domain and range, and find the x-value where f(x) = 0.

  4. Combined domain restrictions. Fluency

    • (a) f(x) = √(x − 1) / (x − 4) — state the natural domain.
    • (b) g(x) = 1/√(x + 2) — state the natural domain.
    • (c) For f(x) = x² with domain restricted to {−2, −1, 0, 1, 2}, state the range.
    • (d) For h(x) = √x, what is the smallest integer in the domain with h(x) > 4?

  5. Domain and range from a graph. Understanding

    The graph shows two functions on the same axes.

    x y −1 1 2 3 −2 1 2 −1 −2 A: y = √x B: y = −x²+2 (0, 2)
    Curve A (blue, closed circle at left, open at right): domain [0, 3). Curve B (red): full parabola.
    • (a) State the domain and range of Curve A (y = √x, drawn from x = 0 to x = 3 with open endpoint at x = 3).
    • (b) State the domain and range of Curve B (y = −x² + 2).
    • (c) Find all x-values where A and B intersect (i.e. √x = −x² + 2).
    • (d) On Curve B, find the values of x for which y ≤ 0.

  6. Restricted domains. Understanding

    • (a) f(x) = x² has domain restricted to x ≥ 0. State the range and describe what changed from the natural domain.
    • (b) g(x) = √(4 − x²) — find the natural domain and range, and name the shape of this graph.
    • (c) A function has domain [−2, 5] and rule f(x) = 2x + 1. Find the range.
    • (d) h(x) = 1/(x² − 4). Find the natural domain.

  7. Real-world domain and range. Understanding

    A ball is thrown upward. Its height (metres) at time t (seconds) is h(t) = −5t² + 20t for 0 ≤ t ≤ 4.

    • (a) State the domain of h in this context.
    • (b) Find the maximum height and when it occurs.
    • (c) State the range of h in this context.
    • (d) For what values of t is h(t) ≥ 15? Write an inequality and solve it.

  8. Comparing function types. Understanding

    • (a) Explain why f(x) = x³ has range = all reals, but g(x) = x² has range y ≥ 0.
    • (b) The function f(x) = a(x − h)² + k has vertex at (h, k). For what values of a does f have range y ≥ k? For what values of a does it have range y ≤ k?
    • (c) A function with range [−3, 7] is reflected in the x-axis. State the new range.
    • (d) f(x) = √x is shifted 4 units right and 3 units down. Write the new function and state its domain and range.

  9. Finding range algebraically. Problem Solving

    • (a) Find the range of f(x) = (x + 2)/(x − 1) by letting y = f(x) and solving for x in terms of y. What y-value is excluded?
    • (b) Find the range of g(x) = 3 − 2x² (no domain restriction).
    • (c) Find the range of h(x) = √(x² − 4) and state any restrictions on x needed for the function to be defined.
    • (d) A quadratic f(x) = ax² + bx + c has range y ≥ −5 and axis of symmetry x = 3. Write a possible equation for f(x).

  10. Domain and range analysis. Problem Solving

    A company's weekly profit (thousand dollars) is modelled by P(n) = −2n² + 80n − 400, where n is the number of units produced (0 ≤ n ≤ 50).

    • (a) State the domain in context.
    • (b) Find the maximum profit and the production level that achieves it.
    • (c) Find the values of n for which the company makes a profit (P > 0).
    • (d) State the range of P for the given domain.