RevisionDojo

SL 2.2—Functions, notation domain, range and inverse as reflection - IB Questionbank | RevisionDojo

Practice SL 2.2—Functions, notation domain, range and inverse as reflection with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

Given the functions $f(x) = 3x - 2$ and $g(x) = \frac{x+2}{3}$ .

Prove that $f(x)$ and $g(x)$ are inverse functions of each other.

[5]

Question 2

SLPaper 1

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f(x)=4 x^{2}-8 x+5$ and $g(x)=x+k$ , where $k \in \mathbb{R}$ . The function $f$ models the height of a projectile in metres over a horizontal distance $x$ in metres.

Find the range of $f$ .

[2]

Given that $(g \circ f)(x) \geq 3$ for all $x \in \mathbb{R}$ , determine the set of possible values for $k$ .

[4]

Question 3

SL & HLPaper 1

Given the functions $f(x) = 2x + 1$ and $g(x) = \frac{x-1}{2}$ ,

Prove that $f(g(x)) = x$ for all real numbers $x$ .

[3]

Question 4

HLPaper 1

Let $h(x) = \frac{1}{x^2}$ and $k(x) = \ln(x)$ .

Find the composite function $(h \circ k)(x)$ .

[1]

Determine the domain of the composite function $(h \circ k)(x)$ with range $y\in\R,y\not=\frac{1}{4}$

[5]

Evaluate the limit $\lim_{x \to 1} (h \circ k)(x)$ .

[2]

Now consider the function $f(x)=k(x)\sqrt{h(x)}$ . Hence, find $\int\! f(x)dx$ .

[5]

Question 5

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+9),\quad x>-9,\ k>0.$ It is given that $(f\circ g)(1)=7$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

Find $(f\circ g)^{-1}(y)$ and state domain/range.

[4]

Question 6

SLPaper 2

A function $f$ is defined by $f(x)=\sqrt{x^{2}+4}-2, x \geq 0$ , representing the path of a river on a map, where both axes represent distance in kilometres.

A function $g$ is defined by $g(x)=e^{-x}+1, x \geq 0$ , representing the path of a railway on the same map.

The origin, $O(0,0)$ , is the location of the centre of a town called Greenville. A straight footpath, $P$ , is built to connect the centre of Greenville to the river at the point where $x=2$ .

Bridges are located where the railway crosses the river. A straight road is built from the centre of Greenville, due east, to connect the town to the railway.

Find the value of $f(x)$ when $x=2$ .

[2]

Find the function $P(x)$ that defines this footpath on the map.

[5]

State the domain of $P$ .

[2]

Find the coordinates of the bridges relative to the centre of Greenville, giving your answers to three significant figures.

[3]

Find the distance from the centre of Greenville to the point at which the road meets the railway.

[2]

The straight road crosses the railway and continues due east. State whether the straight road will ever cross the river. Justify your answer.

[2]

Question 7

SLPaper 2

A function $f$ is defined by

f(x)= \begin{cases}x^{2}-4 & \text { if }-2 \leq x \leq 2 \\ 6-x & \text { if } 2<x \leq 4\end{cases}

representing the height of a sculpture in metres over a horizontal distance $x$ in metres. Figure 1: Graph of $y=f(x)$ .

Find the value of $(f \circ f)(0)$ .

[2]

Given that $f^{-1}(b)=1$ , determine the value of $b$ .

[2]

Given that $g(x)=3 f(x+1)$ , find the domain and range of $g$ .

[3]

Question 8

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+7),\quad x>-7,\ k>0.$ It is given that $(f\circ g)(0)=3$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and state maximal domains.

[4]

Solve $(x+7)^k=\ln(7+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

Find $(f\circ g)^{-1}(y)$ and its domain/range.

[4]

Question 9

SLPaper 2

For a real parameter $p$ , consider the quadratic $q_p(x)=x^2-(p+2)x+(2p-5).$

Find all real values of $p$ for which $q_p$ has two distinct real roots and both roots are greater than $0$ .

[6]

Find the exact roots of $q(x)=0$ .

[3]

Solve $q(x)=g(x)$ for $x\ge0$ to three decimal places. Justify the number of solutions on $x\ge0$ .

[6]

Solve $q(x)\ge g(x)$ for $x\ge0$ (endpoints to 3 d.p.).

[3]

Question 10

SLPaper 2

Consider the equation $e^{0.4x}=\sin x+2$ for $x \geq 0$ .

Use technology to solve the equation for $0\le x\le12$ . Give all solutions correct to 3 d.p.

[4]

By considering the graphs, justify why no further solutions exist for $x>12$ .

[3]

Consider instead $e^{0.4x}=\cos x+3$ . State and justify the number of real solutions for $x\ge0$ .

[4]

SL 2.2—Functions, notation domain, range and inverse as reflection Questionbank