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SL 2.2—Functions, domains, range, graphs - IB Questionbank | RevisionDojo

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

HLPaper 1

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

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 2

SLPaper 2

A function is defined by $f(x) = x + 1$ .

Find the range of $f$ .

[2]

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

[2]

Find the range of $f^{-1}$ .

[2]

Question 3

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 4

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 $10\text{ m}$ 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 has a mass of $50\text{ kg}$ , find the gravitational force component acting along the ramp (using trigonometry).

[3]

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

[3]

Question 5

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 6

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]

Question 7

SLPaper 2

A car's velocity $v(t)$ (in m/s) as a function of time $t$ is modelled by: $v(t) = \frac{120t}{t^2 - 25}$

Find the domain of the function $v(t)$ .

[2]

Explain the physical significance of any restriction in the domain.

[2]

Describe what happens to the speed of the car as $t$ approaches the values where the function is undefined.

[2]

Question 8

SLPaper 2

A local environmental organization is studying the net population change of a particular species, which can be modeled by the quadratic function $f(x) = x^2 + 4x + 3$ . To better understand the population dynamics, they want to analyze the function's behavior and its inverse.

Restrict the domain of the function so that it is one-to-one, allowing for a clearer analysis of the species' population change.

[2]

Find the inverse function $f^{-1}(x)$ to determine the input value that corresponds to a given output in the context of the population change.

[2]

Sketch the graph of $f(x)$ and $f^{-1}(x)$ on the same set of axes, indicating how the population change relates to its inverse function.

[3]

Question 9

SLPaper 2

A company's revenue $R(x)$ (in thousand dollars) from selling $x$ units of a product is modeled by: $R(x) = \frac{1200x}{x + 4}$

Find the range of the revenue function $R(x)$ .

[2]

Discuss the maximum possible revenue as $x$ increases indefinitely.

[2]

What does the lower bound of the range represent in terms of sales?

[2]

Paper

Level

Question 1

HLPaper 1

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

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 2

SLPaper 2

A function is defined by $f(x) = x + 1$ .

Find the range of $f$ .

[2]

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

[2]

Find the range of $f^{-1}$ .

[2]

Question 3

SLPaper 2

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 4

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 $10\text{ m}$ 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 has a mass of $50\text{ kg}$ , find the gravitational force component acting along the ramp (using trigonometry).

[3]

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

[3]

Question 5

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 6

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]

Question 7

SLPaper 2

A car's velocity $v(t)$ (in m/s) as a function of time $t$ is modelled by: $v(t) = \frac{120t}{t^2 - 25}$

Find the domain of the function $v(t)$ .

[2]

Explain the physical significance of any restriction in the domain.

[2]

Describe what happens to the speed of the car as $t$ approaches the values where the function is undefined.

[2]

Question 8

SLPaper 2

Restrict the domain of the function so that it is one-to-one, allowing for a clearer analysis of the species' population change.

[2]

Find the inverse function $f^{-1}(x)$ to determine the input value that corresponds to a given output in the context of the population change.

[2]

Sketch the graph of $f(x)$ and $f^{-1}(x)$ on the same set of axes, indicating how the population change relates to its inverse function.

[3]

Question 9

SLPaper 2

A company's revenue $R(x)$ (in thousand dollars) from selling $x$ units of a product is modeled by: $R(x) = \frac{1200x}{x + 4}$

Find the range of the revenue function $R(x)$ .

[2]

Discuss the maximum possible revenue as $x$ increases indefinitely.

[2]

What does the lower bound of the range represent in terms of sales?

[2]

Paper

Level

Question 1

HLPaper 1

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

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 2

SLPaper 2

A function is defined by $f(x) = x + 1$ .

Find the range of $f$ .

[2]

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

[2]

Find the range of $f^{-1}$ .

[2]

Question 3

SLPaper 2

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 4

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 $10\text{ m}$ 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 has a mass of $50\text{ kg}$ , find the gravitational force component acting along the ramp (using trigonometry).

[3]

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

[3]

Question 5

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 6

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]

Question 7

SLPaper 2

A car's velocity $v(t)$ (in m/s) as a function of time $t$ is modelled by: $v(t) = \frac{120t}{t^2 - 25}$

Find the domain of the function $v(t)$ .

[2]

Explain the physical significance of any restriction in the domain.

[2]

Describe what happens to the speed of the car as $t$ approaches the values where the function is undefined.

[2]

Question 8

SLPaper 2

Restrict the domain of the function so that it is one-to-one, allowing for a clearer analysis of the species' population change.

[2]

Find the inverse function $f^{-1}(x)$ to determine the input value that corresponds to a given output in the context of the population change.

[2]

Sketch the graph of $f(x)$ and $f^{-1}(x)$ on the same set of axes, indicating how the population change relates to its inverse function.

[3]

Question 9

SLPaper 2

A company's revenue $R(x)$ (in thousand dollars) from selling $x$ units of a product is modeled by: $R(x) = \frac{1200x}{x + 4}$

Find the range of the revenue function $R(x)$ .

[2]

Discuss the maximum possible revenue as $x$ increases indefinitely.

[2]

What does the lower bound of the range represent in terms of sales?

[2]

SL 2.2—Functions, domains, range, graphs Questionbank