Functions

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1. 1Composite functions and inversesPut one function inside another, in the right order; find an inverse by making x the subject; and combine the two, as in g⁻¹∘f.Done18 min
2. 2Domain and rangeWhich inputs a function accepts and which outputs it gives: division by zero, square roots and logarithms; reading a range; and why the domain of an inverse is the range of the function.Done16 min

Past questions (22)

From 2008–2023: 20 WAEC, 2 NECO. Theory questions first, then objective, newest first.

Theory (13)

• WAEC 2023 · Paper 2 · Q9Two functions $g$ and $h$ are defined on the set of real numbers, $\mathbb R$, by $g: x \to \dfrac{2 + x}{x^2 + x - 20}$ and $h: x \to x + 3$.
• WAEC 2022 · Paper 2 · Q3Two functions $f$ and $g$ are defined on the set of real numbers, $\mathbb{R}$, by $f : x \to x^2 + 2$ and $g : x \to \dfrac{1}{x + 2}$, $x \ne -2$. …
• WAEC 2017 · Paper 2 · Q1If $f(x) = \dfrac{4 - 5x}{2}$ and $g(x) = x + 6$, $x \in \mathbb R$, find $f \circ g^{-1}$.
• WAEC 2017 · Paper 2 · Q11Find the values of $x$ at the points of intersection of the curves $y = (2x - 5)^3$ and $y = \dfrac{1}{(2x - 5)^3}$.
• WAEC 2016 · Paper 2 · Q3If $f(x + 2) = 6x^2 + 5x - 8$, find $f(5)$.
• WAEC 2016 · Paper 2 · Q2A function $f$ is defined on the set $\mathbb R$ of real numbers by $f : x \to mx^2 + nx + 2$, where $m$ and $n$ are constants. If $f(-2) = 0$ …
• WAEC 2014 · Paper 2 · Q11If $f(x) = \dfrac{x - 3}{2x - 1}$, $x \ne \frac12$ and $g(x) = \dfrac{x - 1}{x + 1}$, $x \ne -1$, find $g \circ f$.
• WAEC 2014 · Paper 2 · Q1Given $f(y) = \log_2\left(\dfrac{y + 1}{y + 3}\right)$:
• WAEC 2011 · Paper 2 · Q10If $\mathbf A = \begin{pmatrix} -2 & 5 \\ 4 & 3 \end{pmatrix}$ and $\mathbf B = \begin{pmatrix} 3 & 1 \\ 2 & 3 \end{pmatrix}$, find the values …
• WAEC 2011 · Paper 2 · Q9Use the trapezium rule with ordinates at $x = 1, 2, 3, 4$ and $5$ to calculate, correct to two decimal places, an approximate value for $\displaystyle\int_1^5 (2x + 8x^{-2})\,dx$.
• WAEC 2009 · Paper 2 · Q2Given that $f(x) = \dfrac{x}{2}$, $g(x) = \dfrac{5x}{x^2 - 4}$, $x \ne \pm 2$, and $h(x) = (x - 1)^{\frac12}$, where $x \in \mathbb R$, the set …
• WAEC 2008 · Paper 2 · Q1A function $f$ is defined on the set $\mathbb{R}$ of real numbers by $f : x \to px^2 + qx + 2$, where $p$ and $q$ are constants. If $f(-2) = 0$ …
• WAEC 2008 · Paper 2 · Q1Two functions $f$ and $g$ are defined on the set $\mathbb{R}$ of real numbers by $f : x \to \dfrac{x + 1}{x - 2}$, $x \neq 2$, and $g : x \to \dfrac{kx + 3}{x}$, …

Objective (9)

• NECO 2023 · Paper 1 · Q19Let $f$ and $g$ on the set of real numbers be defined by $f(x) = 3x^2 - 4$ and $g(x) = 2x + 1$. Find $fg(x)$.
• NECO 2023 · Paper 1 · Q25Find the values of $x$ for which $g(x) = \dfrac{x^2 - 5x + 6}{x^2 - 5x - 24}$ is undefined.
• WAEC 2023 · Paper 1 · Q11A function $f$ is defined by $f: x \rightarrow \dfrac{x + 2}{x - 3}$, $x \neq 3$. Find the inverse of $f$.
• WAEC 2023 · Paper 1 · Q9If $f: x \rightarrow 2\tan x$ and $g: x \rightarrow \sqrt{x^2 + 8}$, find $(g \circ f)(45^\circ)$.
• WAEC 2022 · Paper 1 · Q24If $g(x) = \sqrt{1 - x^2}$, find the domain of $g(x)$.
• WAEC 2022 · Paper 1 · Q39If $f(x - 1) = x^3 + 3x^2 + 4x - 5$, find $f(2)$.
• WAEC 2022 · Paper 1 · Q6The functions $f: x \to 2x^2 + 3x - 7$ and $g: x \to 5x^2 + 7x - 6$ are defined on the set of real numbers, $R$. Find the values of $x$ for which …
• WAEC 2020 · Paper 1 · Q11Given that $f(x) = 2x$ and $g(x) = 3x^2$, find the values of $x$ for which $f^{-1}(x) = g(x)$.
• WAEC 2020 · Paper 1 · Q6Given that $x : \mathbb{R} \to \mathbb{R}$ is defined by $x = \dfrac{y + 1}{5 - y}$, $y \in \mathbb{R}$, find the domain of $x$.