AHL 2.16—Graphing modulus equations and inequalities Questionbank

Practice AHL 2.16—Graphing modulus equations and inequalities 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

HLPaper 1

Consider the functions $f(x)=|\ln | \frac{x}{x-k}| |, x \in \mathbb{R} \backslash\{0, k\}$ , and $g(x)=\arctan (|x-2 k|)$ , $x \in \mathbb{R}$ , where $k \in \mathbb{R}, k>3$ . The graphs of $y=f(x)$ and $y=g(x)$ intersect at a point $P$ .

Describe the transformation that maps the graph of $y=\ln \left|\frac{x}{x-1}\right|$ to $y=\ln \left|\frac{x}{x-k}\right|$ .

[2]

State the range of $f$ .

[2]

On the set of axes below, sketch the graphs of $y=f(x)$ and $y=g(x)$ . Clearly indicate any asymptotes, points of intersection with the axes, and the coordinates of point $P$ .

[6]

Find the coordinates of $P$ .

[3]

Find $\int f(x) \sin x d x$ .

[4]

Question 2

HLPaper 2

Let $f(x)=\left|x^{2}-2 k x+k^{2}\right|, x \in \mathbb{R}$ , and $g(x)=\frac{1}{\left|\frac{x-k}{x+k}\right|+1}, x \in \mathbb{R} \backslash\{-k\}$ , where $k>2$ . The function $h$ is defined as $h(x)=\arctan (f(x))$ .

Express $f(x)$ in the form

f(x)= \begin{cases}a x^{2}+b x+c & \text { if } x<k \\ d & \text { if } x=k \\ e x^{2}+f x+g & \text { if } x>k\end{cases}

[3]

Find the equations of the vertical and horizontal asymptotes of $y=g(x)$ .

[2]

On the set of axes below, sketch the graph of $y=g(x)+h(x)$ . Clearly indicate any asymptotes, intercepts with the axes, and any local minima or maxima.

[5]

Find the range of $y=g(x)+h(x)$ .

[3]

Solve the inequality $g(x)+h(x) \leq 1$ . Give your answer in interval notation.

[3]

Question 3

HLPaper 1

The diagram below shows the graph of $y=h(x)$ , where $h(x)=x-1$ . The graph crosses the $x$ -axis at $x=1$ and the $y$ -axis at $y=-1$ .

On the same set of axes, sketch the graph of $y=|h(x)|-1$ , clearly showing the coordinates of any intercepts and vertices.

[3]

Solve the equation $|h(x)|=1$ , giving exact solutions.

[2]

Question 4

HLPaper 1

Let $f(x)=|2 x+1|-3, x \in \mathbb{R}$ .

Sketch the graph of $y=f(x)$ on the axes below. Clearly indicate any intercepts with the axes and the vertex of the graph.

[3]

Write down the range of $f$ .

[2]

Express $f(x)$ in the form

f(x)= \begin{cases}a x+b & \text { if } x<-\frac{1}{2} \\ c & \text { if } x=-\frac{1}{2} \\ d x+e & \text { if } x>-\frac{1}{2}\end{cases}

[3]

Solve the inequality $f(x) \geq-1$ .

[2]

Question 5

HLPaper 1

Let $f(x)=\left|\frac{2 x+1}{x-3}\right|, x \in \mathbb{R} \backslash\{3\}$ , and $g(x)=\arctan (|x-1|), x \in \mathbb{R}$ . The inverse of $f$ restricted to $x>3$ is $f^{-1}$ .

Find the equations of the vertical and horizontal asymptotes of $y=f(x)$ .

[2]

On the set of axes below, sketch the graph of $y=f(x)$ . Clearly indicate the asymptotes, any points of intersection with the axes, and any local minima or maxima.

[4]

Show that $f^{-1}(x)=\frac{3 x+1}{x+2}$ for $x>0$ .

[4]

Solve the inequality $f(x) \leq g(x)$ . Give your answer in interval notation.

[3]

Question 6

HLPaper 2

Let $f(x)=\left|x^{2}-4 x+3\right|, x \in \mathbb{R}$ , and $g(x)=\frac{2}{|x-2|+1}, x \in \mathbb{R}$ .

Express $f(x)$ in the form

f(x)= \begin{cases}a x^{2}+b x+c & \text { if } x<1 \\ d & \text { if } x=1 \\ e x^{2}+f x+g & \text { if } 1<x<3 \\ h x^{2}+i x+j & \text { if } x>3\end{cases}

[4]

On the set of axes below, sketch the graph of $y=f(x)+g(x)$ . Clearly indicate any asymptotes, intercepts with the axes, and any local minima or maxima.

[5]

Find the range of $y=f(x)+g(x)$ .

[3]

Solve the inequality $f(x)+g(x)<2$ . Give your answer in interval notation.

[3]

Question 7

HLPaper 2

A function $g$ is defined by $g(x)=\frac{2 x-3}{x+1}$ , where $x \in \mathbb{R}, x \neq-1$ .

Find the equations of the vertical and horizontal asymptotes of the graph of $y=$ $g(x)$ .

[2]

On the set of axes below, sketch the graph of $y=|g(x)|+1$ . Clearly indicate the asymptotes, any points of intersection with the axes, and any local minima or maxima.

[3]

Solve the inequality $|g(x)|+1 \geq 3$ . Give your answer in interval notation.

[2]

Question 8

HLPaper 1

The diagram below shows the graph of $y=k(x)$ , where $k(x)=-x+3$ . The graph crosses the $x$ -axis at $x=3$ and the $y$ -axis at $y=3$ .

On the same set of axes, sketch the graph of $y=2|k(x)|$ , clearly showing the coordinates of any intercepts and vertices.

[3]

Solve the inequality $2|k(x)|<6$ , giving your answer in interval notation.

[2]

Question 9

HLPaper 1

Consider the functions $f(x)=\ln |x-1|, x \in \mathbb{R} \backslash\{1\}$ , and $g(x)=\ln |x-k|, x \in \mathbb{R} \backslash\{k\}$ , where $k \in \mathbb{R}, k>3$ .

The graphs of $y=f(x)$ and $y=g(x)$ intersect at the point $P$ .

Describe the transformation that maps the graph of $y=f(x)$ to $y=g(x)$ .

[1]

State the range of $g$ .

[1]

On the set of axes below, sketch the graphs of $y=f(x)$ and $y=g(x)$ . Clearly indicate the asymptotes and the coordinates of any points of intersection with the axes.

[4]

Find the coordinates of the point $P$ .

[2]

Question 10

HLPaper 1

The diagram below shows the graph of $y=f(x)$ , where $f(x)=x+2$ . The graph crosses the $x$ -axis at $x=-2$ and the $y$ -axis at $y=2$ .

On the same set of axes, sketch the graph of $y=|f(x)|$ , clearly showing the coordinates of any intercepts and vertices.

[3]

Solve the inequality $|f(x)| \leq 2$ , giving your answer in interval notation.

[2]