AHL 2.7—Composite functions, finding inverse function incl domain restriction Questionbank

Practice AHL 2.7—Composite functions, finding inverse function incl domain restriction with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A local health organization is analyzing the function ${g(x) = 3 \ln x + \frac{2}{x-2}}$ , which models the relationship between the amount of nutrients in a food source and its health benefits, defined for ${0 < x < 4}$ .

Find the range of ${g(x)}$ .

[3]

Determine the coordinates of the point of inflection on the graph of ${y = g(x)}$ given that ${g(x) = 3 \ln x + \frac{2}{x-2}}$

[2]

Sketch the graph of ${y = g(x)}$ , showing clearly any axis intercepts and giving the equations of asymptotes.

[Graph shows a rational function with a vertical asymptote at x = 2, horizontal asymptote at y = 1, and x-intercept at (0,0)]

[4]

Question 2

HLPaper 2

A biotechnology firm is studying the effects of a new drug, modeled by the function ${g(x) = 2x \arcsin(x)}$ for ${-1 \leq x \leq 1}$ .

Sketch the graph of ${g(x)}$ , indicating intercepts and local extrema.

[3]

State the range of ${g}$ .

[2]

Solve the inequality ${2x \arcsin(x) > 1}$ .

[3]

Question 3

HLPaper 1

The function $f$ is defined by $f(x) = 2{x^3} + 5,{\text{ }} - 2 \leqslant x \leqslant 2$ .

Write down the range of $f$ .

[2]

Find an expression for ${f^{ - 1}}(x)$ .

[2]

Write down the domain and range of ${f^{ - 1}}$ .

[2]

Question 4

HLPaper 2

The function $f(x)=\ln\left(\frac{1}{x-2}\right)$ is defined for $x>2, x \in \mathbb{R}$ .

Find an expression for $f^{-1}(x)$ . You are not required to state a domain.

[4]

Solve $f(x)=f^{-1}(x)$ .

[1]

Question 5

HLPaper 2

A researcher is studying the growth of a specific plant species, modeled by the function ${h(x) = 2 + \ln(x^2 - 4)}$ where ${x \in D}$ .

Find the largest possible domain ${D}$ for ${h}$ to be a function.

[2]

Sketch the graph of ${y = h(x)}$ , showing the equations of asymptotes and coordinates of intercepts.

[3]

Explain why ${h}$ is an even function.

[3]

Explain why the inverse function ${h^{-1}}$ does not exist.

[1]

Question 6

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow. The path of the ball is modelled by the equation $\binom{x}{y} = \binom{5}{0} + t \binom{u_x}{u_y - 5t}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$

Find the initial speed of the ball.

[2]

Find the angle of elevation of the ball as it is launched.

[2]

Find the maximum height reached by the ball.

[3]

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

An archer releases an arrow from the point (0, 2). The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed 60 m s⁻¹ and an angle of elevation of 10°. Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

[4]

Question 7

SLPaper 2

Let $f(x) = {x^2} + 2x + 1$ and $g(x) = x - 5$ , for $x \in \mathbb{R}$ .

Find $(g \circ f)(x)$ .

[2]

Find $f(8)$ .

[2]

Solve $(g \circ f)(x) = 0$ .

[3]

Question 8

SLPaper 1

Let $f(x) = 1 + {{\text{e}}^{ - x}}$ and $g(x) = 2x + b$ , for $x \in \mathbb{R}$ , where $b$ is a constant.

Find $(g \circ f)(x)$ .

[2]

Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .

[4]

Question 9

HLPaper 2

The function $f$ is given by $f(x) = m{x^3} + n{x^2} + px + q$ , where $m$ , $n$ , $p$ , $q$ are integers.

The graph of $f$ passes through the point (0, 0).

The graph of $f$ also passes through the point (3, 18).

The graph of $f$ also passes through the points (1, 0) and (–1, –10).

Write down the value of $q$ .

[1]

Show that $27m + 9n + 3p = 18$ .

[2]

Write down the other two linear equations in $m$ , $n$ and $p$ .

[2]

Write down these three equations as a matrix equation.

[3]

Solve this matrix equation.

[3]

The function $f$ can also be written $f(x) = x\left( {x - 1} \right)\left( {rx - s} \right)$ where $r$ and $s$ are integers. Find $r$ and $s$ .

[3]

Question 10

HLPaper 2

A renewable energy company is analyzing the function ${f(x) = \frac{16}{x} + x^2 - 8}$ for ${x > 0}$ to optimize their solar panel design.

State the equation of the vertical asymptote of ${f(x)}$ .

[1]

Find ${f'(x)}$ where ${f(x) = \frac{16}{x} + \frac{x^2}{8}}$

[3]

Write down the interval in which ${f(x)}$ is increasing.

[2]

$u_x$ is the horizontal component of the initial velocity
$u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$