Functions: Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Functions revision notes

---

Section A

1. Explain what f(x) means and what is meant by the domain and range of a function. [3]

2. For f(x) = 3x − 5, find f(4) and f(−2). [2]

Section B

3. f(x) = 2x + 1 and g(x) = x².

(a) Find fg(3). [2] (b) Find gf(3). [2] (c) Explain why fg(x) and gf(x) are generally different. [2] (d) Find an expression for fg(x) in terms of x. [2]

4. f(x) = (x + 4)/3.

(a) Find f⁻¹(x). [3] (b) Verify that ff⁻¹(2) = 2. [2] (c) Explain the geometrical relationship between the graphs of f(x) and f⁻¹(x). [2]

5. g(x) = 1/(x − 2).

(a) State the value of x that must be excluded from the domain and explain why. [2] (b) Find g⁻¹(x). [3]

6. f(x) = x² − 6x + 11 for x ≥ 3.

(a) Write it in completed square form. [2] (b) State the range of f. [2] (c) Explain why the domain is restricted to x ≥ 3 for the inverse to exist. [3]

7. h(x) = √(x − 3).

(a) State the values of x that must be excluded from the domain, and explain why. [2] (b) State the domain of h using inequality notation. [1]

8. Using a mapping diagram to explain your reasoning, distinguish a one-to-one function from a many-to-one function, and state which type has an inverse function without needing a restricted domain. [3]

---

Answers

1. f(x) means the output of the function f when the input is x [1]. The domain is the set of all permitted inputs [1]; the range is the set of all resulting outputs [1].

2. f(4) = 12 − 5 = 7 [1]; f(−2) = −6 − 5 = −11 [1].

3. (a) g(3) = 9 [1]; f(9) = 19 [1]. (b) f(3) = 7 [1]; g(7) = 49 [1]. (c) The order of operations differs — fg means apply g first, then f, while gf reverses this [1]; since the two operations are not commutative, the results differ except in special cases [1]. (d) fg(x) = f(x²) [1] = 2x² + 1 [1].

4. (a) Let y = (x + 4)/3 [1]; 3y = x + 4; x = 3y − 4 [1]; f⁻¹(x) = 3x − 4 [1]. (b) f⁻¹(2) = 2 [1]; f(2) = 6/3 = 2 ✓ [1]. (c) They are reflections of one another in the line y = x [1], because the inverse swaps the roles of input and output [1].

5. (a) x = 2 must be excluded [1], because it makes the denominator zero and division by zero is undefined [1]. (b) y = 1/(x − 2) [1]; y(x − 2) = 1; x − 2 = 1/y [1]; g⁻¹(x) = 1/x + 2 [1].

6. (a) x² − 6x + 11 = (x − 3)² − 9 + 11 [1] = (x − 3)² + 2 [1]. (b) For x ≥ 3, (x − 3)² ≥ 0 [1], so the range is f(x) ≥ 2 [1]. (c) Without restriction the function is not one-to-one — for example f(2) = f(4) = 3 [1]. An inverse requires each output to come from exactly one input [1]; restricting the domain to x ≥ 3 takes only the right-hand half of the parabola, from the vertex onwards, which is one-to-one [1].

7. (a) Values of x for which x − 3 < 0, i.e. x < 3, must be excluded [1], because the square root of a negative number is not a real value — there is no real number that, squared, gives a negative result [1]. (b) x ≥ 3 [1].

8. In a mapping diagram, a one-to-one function has exactly one arrow arriving at each range value [1]; a many-to-one function has two or more arrows arriving at the same range value [1] (for example, a quadratic maps both x = 2 and x = −2 to the same output, since (2)² = (−2)² = 4). Only a one-to-one function has an inverse that is itself a function without restricting the domain, since a many-to-one function’s reverse mapping would send one input to more than one output, which an inverse function cannot do [1].

---

Where marks are usually lost

• Applying fg in the wrong order — the function nearest x acts first.
• Forgetting to exclude values that make a denominator zero.
• Stating a range as a domain.
• Not explaining why a domain restriction is needed for an inverse.
• Forgetting to exclude values that make the expression under a square root negative, not just values that make a denominator zero.
• Confusing one-to-one with many-to-one when explaining why a function does or does not have an inverse without a restricted domain.

Work through the Functions revision notes alongside these questions: the notes summarise domain, range and the composite/inverse function methods in condensed form, while these questions test whether you can apply the domain-restriction rules to a square-root function and reason about one-to-one versus many-to-one mappings, rather than just recall the definitions.