Functions: 19 IB Maths AI HL practice questions

Consider the following function, $f(x)$, defined on the domain of integers from 0 to 4 inclusive.

(a) Solve $f(x) = 4$.[1]

(b) Solve $f(x) = x$.[1]

(c) Complete the following table.[3]

Professor Wei observed that students have difficulty remembering the information presented in his lectures.

He modelled the percentage of information retained, $R$, by the function $R(t) = 100 e^{-pt}$, $t \geq 0$, where $t$ is the number of days after the lecture.

He found that 1 day after a lecture, students had forgotten 50% of the information presented.

(a) Find the value of $p$.[2]

(b) Use this model to find the percentage of information retained by his students 36 hours after Professor Wei's lecture.[2]

Based on his model, Professor Wei believes that his students will always retain some information from his lecture.

(c) State a mathematical reason why Professor Wei might believe this.[1]

(d) Write down one possible limitation of the domain of the model.[1]

Stars are classified by their brightness. The brightest stars in the sky have a magnitude of 1. The magnitude, $m$, of another star can be modelled as a function of its brightness, $b$, relative to a star of magnitude 1, as shown by the following equation.

$$m = 1 - 2.5 \log_{10}(b)$$

The star called Acubens has a brightness of 0.0525.

(a) Find the magnitude of Acubens.[2]

Ceres has a magnitude of 7 and is the least bright star visible without magnification.

(b) Find the brightness of Ceres.[2]

The star Proxima Centauri has a greater magnitude than the planet Neptune. The difference in their magnitudes is 3.2.

(c) Find how many times brighter Neptune is compared to Proxima Centauri.[3]

Radioactive carbon is a material that decays over time.

The mass, $m(t)$ (in nanograms), of radioactive carbon in a fossil of a plant, after $t$ years, can be modelled by the function

$$m(t) = 120e^{-0.00012t}$$

where $t$ is the time since the plant died.

(a) Write down the initial mass of the radioactive carbon.[1]

(b) Find the mass of the radioactive carbon after 20000 years.[2]

(c) Calculate the smallest number of complete years it takes for more than half the sample to decay.[3]

The height of a baseball after it is hit by a bat is modelled by the function

$$h(t) = -4.8t^2 + 21t + 1.2$$

where $h(t)$ is the height in metres above the ground and $t$ is the time in seconds after the ball was hit.

(a) Write down the height of the ball above the ground at the instant it is hit by the bat.[1]

(b) Find the value of $t$ when the ball hits the ground.[2]

(c) State an appropriate domain for $t$ in this model.[2]

Celeste heated a cup of coffee and then let it cool to room temperature. Celeste found the coffee's temperature, $T$, measured in $°C$, could be modelled by the following function,

$$T(t) = 71e^{-0.0514t} + 23, t \geq 0,$$

where $t$ is the time, in minutes, after the coffee started to cool.

(a) Find the coffee's temperature 16 minutes after it started to cool.[2]

(b) Write down the room temperature.[1]

(c) Given that $T^{-1}(50) = k$, find the value of $k$.[2]

Walkea Tree is a furniture store. They offer a 30% discount on all chairs priced over 250 Canadian dollars (CAD).

A different furniture store, LuxeCraft, offer customers a reduction of 150 CAD on all chairs priced over 250 CAD.

(a)(i) write down a function $f(x)$ that models the final price of a chair at Walkea Tree, after the discount is applied.[1]

(a)(ii) write down a function $g(x)$ that models the final price of a chair at LuxeCraft, after the reduction is applied.[2]

A third store, Opulent Furnishings, decides to offer both a 30% discount and a reduction of 150 CAD on all their chairs priced over 250 CAD.

(b) Describe the meaning of $(f \circ g)(x)$, in context.[2]

(c) State, with justification, whether $(f \circ g)(x)$ or $(g \circ f)(x)$ is better for the customer when they buy a chair with an original price of 450 CAD.[2]

Roma is told by the manufacturers of their phone that, when charging from $0\%$, the percentage $P$ of charge can be modelled by $P = 100 - 100 \times 2^{-t}$, where $t$ is the number of hours since the phone started charging.

(a) Find the time taken for the amount of charge to go from $0\%$ to $40\%$, according to this model.[2]

(b) Find the sum of the square residuals ($SS_{res}$) when using $P = 100 - 100 \times 2^{-t}$ to fit this data.[4]

Roma now uses a quadratic function to model the charge in the phone. They calculate that, when using this model, the sum of the square residuals for these four points is $2.45$.

(c) State one reason why Roma might prefer to use

(i) the quadratic model.[2]

(ii) $P = 100 - 100 \times 2^{-t}$.[2]

Consider the function $B(x) = 20 + \frac{32}{x} + 2x$, where $x \geq 1$, $x \in \mathbb{R}$.

(a)(i) Find $B'(x)$.[2]

(a)(ii) Use your answer to part (a)(i) to find the minimum value of $B(x)$. You may assume that $B(x)$ has no local maximum point over the given domain.[4]

Kim goes to a restaurant that offers a large family banquet. The time, $T$, in minutes, to prepare the family banquet depends on the number of chefs, $n$.

To predict the value of $T$, Kim uses the model $T(n) = 20 + \frac{32}{n} + 2n$, where $n \in \mathbb{Z}^+$.

The restaurant informs Kim that the time taken to prepare the family banquet is less than 40 minutes. There are $k$ chefs preparing the family banquet.

(b) Find the possible values of $k$.[2]

The graph of the function $f(x) = \ln x$ is translated by $\begin{pmatrix} a \\ b \end{pmatrix}$ so that it then passes through the points $(0, 1)$ and $(e^3, 1 + \ln 2)$.

Find the value of $a$ and the value of $b$.[7]

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

(a) Find an expression for $f^{-1}(x)$. You are not required to state a domain.[3]

(b) Solve $f(x) = f^{-1}(x)$.[2]

A cup of hot water is placed in a room and is left to cool for half an hour. Its temperature, measured in $°C$, is recorded every 5 minutes. The results are shown in the table.

Akira uses the power function $T(t) = at^b + 25$ to model the temperature, $T$, of the water $t$ minutes after it was placed in the room.

(a) State what the value of 25 represents in this context.[1]

(b) Use your graphic display calculator to find the value of $a$ and of $b$.[3]

Soo Min models the temperature, $T$, of the water $t$ minutes after it was placed in the room as $T(t) = kc^t + 25$.

(c) Find the value of $k$ and of $c$.[1]

(d) State a reason why Soo Min's model of the temperature is a better fit for the data than Akira's model.[1]

(a)(i) Write down the value of $P(0)$.[1]

(a)(ii) Interpret what this value means in this context.[2]

(b) Find the value of $k$.[2]

(c) Find the population of the bacteria 1 hour and 30 minutes after the experiment began.[2]

(d) Find the value of $t$ when the two populations of bacteria are equal.[2]

(e) Find the value of $m$, giving your answer as an integer value.[4]

A function is defined by $f(x) = 2 - \frac{12}{x+5}$ for $-7 \le x \le 7, x \ne -5$.

(a) Find the range of $f$.[3]

(b) Find an expression for the inverse function $f^{-1}(x)$. The domain is not required.[3]

(c) Write down the range of $f^{-1}(x)$.[1]

Ethan and Avery are researching population data for the city of Los Angeles to create a model predicting population values. They collect the following data.

Ethan proposes the population can be modelled using quadratic regression to find a function of the form $f(x) = ax^2 + bx + c$, where $x$ is the number of years after 1900.

(a) Find the equation of Ethan's model.[3]

Ethan finds the coefficient of determination for his model is 0.98843 to five significant figures.

(b) State whether the coefficient of determination supports Ethan's proposal. Justify your answer.[2]

(c) Comment on the validity of Ethan's model with reference to one of the parameters in the equation.[1]

(d) (i) Find the value of $f'(110)$ and interpret this value in context.[3]

(ii) By considering the population changes in the table, use the value found in part (d)(i) to comment on the validity of Ethan's model.[4]

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Avery proposes that the population instead follows a logistic model of the form

$$g(x) = \frac{4000}{1 + 14e^{-0.05x}}$$

where $x$ is the number of years after 1900.

(e) State a reason why it may be valid to use Avery's proposal to predict the future population.[1]

(f) (i) Find $g'(x)$.[3]

(ii) Hence find the year, according to Avery's model, during which the greatest population growth rate occurred.[6]