Functions — AS & A Level Mathematics (9709) questions by topic
Deriving an inverse function from a restricted domain, reading off two piecewise-linear graphs plotted on the same grid, completing the square on a quadratic expression, and using a function's range to pin down an unknown constant are typical of the 18 functions questions from 2023 to 2025, worth 4 to 13 marks.
Find, state and describe lead the command words, with sketch and solve used where a graph must be drawn or an equation involving a composite function solved. Because domain and range restrictions are the most common source of lost marks here, submit your inverse-function and range working for instant marking, so an ignored restriction or a wrongly swapped x and y is caught before it invalidates the rest of the answer.
May–June 2025, Paper 11
The functions f and g are defined by f(x) = √x for x ⩾ 0, g(x) = 3√(x + 2) − 5 for x ⩾ −2. Describe fully a sequence of transformations which transforms the graph of y = f(x) to…
Answer this question and get it marked →May–June 2025, Paper 12
The diagram shows the graphs with equations y = f(x) and y = g(x). The graph of y = f(x): starts at (−2, 4), rises to a maximum at (0, 6), then falls linearly to (3, 3) and…
Answer this question and get it marked →Express x^2 + 4x + 2 in the form (x + a)^2 + b, where a and b are integers.
Answer this question and get it marked →May–June 2025, Paper 13
The function f is defined by f(x) = x² + 4ax + a for x ∈ ℝ, where a is a constant. The function g is such that g⁻¹(x) = ³√(2x − 4) for x ∈ ℝ. Given that the range of f is f(x) ⩾…
Answer this question and get it marked →May–June 2025, Paper 15
Functions f and g are defined as follows. f(x) = cos x for 0 ⩽ x ⩽ π g(x) = 3 cos(x − π) + 2 for π ⩽ x ⩽ 2π Describe fully the transformations that have been combined to transform…
Answer this question and get it marked →May–June 2024, Paper 12
The curve y = x² is transformed to the curve y = 4(x − 3)² − 8. Describe fully a sequence of transformations that have been combined, making clear the order in which the…
Answer this question and get it marked →The function f is defined by f(x) = √x − 1 for x 1. Find an expression for f⁻¹(x).
Answer this question and get it marked →May–June 2024, Paper 13
The function f is defined by f(x) = 10 + 6x − x^2 for x ∈ ℝ. By completing the square, find the range of f.
Answer this question and get it marked →October–November 2024, Paper 11
The function f is defined by f(x) = 3 + 6x − 2x² for x ∈ ℝ. Express f(x) in the form a − b(x − c)², where a, b and c are constants, and state the range of f.
Answer this question and get it marked →October–November 2024, Paper 12
The function f is defined by f(x) = (2x + 1)/(2x − 1) for x < ½. State the value of f(−1).
Answer this question and get it marked →May–June 2023, Paper 11
The diagram shows two piecewise-linear graphs drawn on a grid (x from about −8 to 6, y from about −6 to 6), with equations y = f(x) and y = g(x). The graph y = f(x) lies mainly in…
Answer this question and get it marked →The functions f and g are defined by f(x) = 1 + 2a/(x − a) for x a and g(x) = bx − 2 for x ∈ ℝ, where a and b are constants. Given that f(7) = 5/2 and gf(5) = 4, find the values…
Answer this question and get it marked →May–June 2023, Paper 12
The diagram shows the graph of y = f(x), where the function f is defined by f(x) = 3 + 2 sin((1/4)x) for 0 ≤ x ≤ 2π. The grid has the x-axis marked at (1/2)π, π, (3/2)π, 2π,…
Answer this question and get it marked →May–June 2023, Paper 13
The diagram shows the graph of y = f(x), which consists of the two straight lines AB and BC, where A is at (1, 3), B is at (2, 5) and C is at (4, 4). The lines A′B′ and B′C′ form…
Answer this question and get it marked →The function f is defined by f(x) = 2 − 5/(x + 2) for x −2. State the range of f.
Answer this question and get it marked →October–November 2023, Paper 11
The transformation R denotes a reflection in the x-axis and the transformation T denotes a translation of the vector (3, −1). Find the equation, y = g(x), of the curve with…
Answer this question and get it marked →October–November 2023, Paper 12
Functions f and g are defined by f(x) = (x + a)² − a for x ≤ −a, g(x) = 2x − 1 for x ∈ ℝ, where a is a positive constant. Find an expression for f⁻¹(x).
Answer this question and get it marked →October–November 2023, Paper 13
The function f is defined by f(x) = 1 + 3/(x − 2) for x 2. State the range of f.
Answer this question and get it marked →Other AS & A Level Mathematics topics
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