SL 2.2—Functions, domains, range, graphs Questionbank

Practice SL 2.2—Functions, domains, range, graphs 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

SLPaper 2

The rate of a chemical reaction ${r(T)}$ , depending on temperature ${T}$ (in degrees Celsius), is modeled by: $r(T) = \frac{1000}{T - 5}$

Find the range of the reaction rate ${r(T)}$ .

[2]

Describe what happens to the reaction rate as the temperature increases indefinitely.

[2]

Question 2

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 3

SLPaper 1

The strength of earthquakes is measured on the Richter magnitude scale, with valuestypically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$ ,which have a magnitude of at least $M$ . For a particular region the equation is

$\log_{10} N = a - M$ , for some $a \in ℝ$ .

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$ .

The equation for this region can also be written as $N = \frac{b}{10^{M}}$ .

The expected length of time, in years, between earthquakes with a magnitude of atleast $M$ is $\frac{1}{N}$ .

Within this region the most severe earthquake recorded had a magnitude of $7.2$ .

Find the value of $a$ .

[2]

Find the value of $b$ .

[2]

Given $0 < M < 8$ , find the range for $N$ .

[2]

Find the expected length of time between this earthquake and the next earthquake ofat least this magnitude. Give your answer to the nearest year.

[2]

Question 4

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 5

SLPaper 2

A new app is designed to track user engagement over time, defined by the function ${f(x) = x + 1}$ .

Find the range of ${f}$ .

[2]

Find an expression for the inverse function ${f^{-1}(x)}$ . The domain is not required.

[2]

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

[2]

Question 6

SLPaper 2

A technology company is developing a new software that predicts user engagement based on a cubic function, modeled by $f(x) = x^3 - 2$ . They are interested in finding the inverse of this function to understand how changes in engagement levels affect user input.

Find the inverse function $f^{-1}(x)$ to determine the input value that corresponds to a specific engagement level.

[2]

Sketch the graph of $f(x)$ , $f^{-1}(x)$ , and the line $y = x$ on the same axes to visualize the relationship between user engagement and input.

[2]

Question 7

SLPaper 1

The graph of a quadratic function has $y$ -intercept 10 and one of its $x$ -intercepts is 1.

The $x$ -coordinate of the vertex of the graph is 3.

The equation of the quadratic function is in the form $y = a{x^2} + bx + c$ .

Write down the value of $c$ .

[1]

Find the value of $a$ and of $b$ .

[4]

Write down the second $x$ -intercept of the function.

[1]

Question 8

SLPaper 2

A delivery service is testing the efficiency of a ramp for loading packages onto delivery trucks. The ramp is inclined at an angle of ${35^\circ}$ and is 10 meters long.

Draw a diagram of the ramp showing the slope and label the vertical rise and horizontal run.

[2]

Calculate the height of the ramp at its highest point.

[2]

If a package weighs 50 kg, find the gravitational force component acting along the ramp (using trigonometry).

[3]

If friction is introduced on the ramp, illustrate how the angle of the slope affects the force needed to move the package upward.

[3]

Question 9

SLPaper 2

A startup's profit ${P(x)}$ (in thousands of dollars) for selling ${x}$ units of a product is given by: $P(x) = \frac{500x}{x^2 + 16}$

Find the domain of the profit function.

[2]

Discuss any real-world restrictions on the number of units ${x}$ that can be sold.

[2]

Analyze the behavior of ${P(x)}$ as ${x \to 0}$ and as ${x \to \infty}$ .

[2]

Question 10

SLPaper 2

Consider the function ${f(x) = 2x + 3}$ where $x \in \mathbb{R}$ .

Find the inverse function ${f^{-1}(x)}$ .

[2]

Plot both ${f(x)}$ and ${f^{-1}(x)}$ on the same set of axes. Use a scale of 1 unit on each axis.

[2]

Draw the line ${y = x}$ on the graph and explain how ${f(x)}$ and ${f^{-1}(x)}$ are related to this line.

[3]

The strength of earthquakes is measured on the Richter magnitude scale, with valuestypically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$ ,which have a magnitude of at least $M$ . For a particular region the equation is

$\log_{10} N = a - M$ , for some $a \in ℝ$ .

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$ .

The equation for this region can also be written as $N = \frac{b}{10^{M}}$ .

The expected length of time, in years, between earthquakes with a magnitude of atleast $M$ is $\frac{1}{N}$ .

Within this region the most severe earthquake recorded had a magnitude of $7.2$ .