Functions: 38 IB Maths AI SL 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]

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]

The graph of $T$ has a horizontal asymptote.

(b) Write down the equation of the horizontal asymptote.[1]

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

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

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 $y = ax^2 + bx + c$.

(a) Write down the value of $c$.[1]

(b) Find the value of $a$ and of $b$.[4]

(c) Write down the second $x$-intercept of the function.[1]

Consider the graph of the function $f(x) = x^2 - \frac{k}{x}$.[3]

(a) Write down $f'(x)$.[3]

The equation of the tangent to the graph of $y = f(x)$ at $x = -2$ is $2y = 4 - 5x$.

(b) Write down the gradient of this tangent.[1]

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

Consider the graph of the following function:

$$g(x) = \frac{8}{x} + \frac{x^2}{2}, \text{ for } x \neq 0.$$

(a) Write down the equation of the vertical asymptote of $g(x)$.[1]

(b) Find $g'(x)$.[3]

(c) Write down the interval in which $g(x)$ is increasing.[2]

M-Line is a company that prints and sells custom designs on T-shirts. For each order, they charge an initial design fee and then an additional fee for each printed T-shirt.

M-Line charges $M$ euros per order. This charge is modelled by the linear function $M(x) = 5x + 40$, where $x$ is the number of T-shirts in the order.

(a) Write down the initial design fee charged for each order.[1]

(b) Find the total amount charged for an order of 94 T-shirts.[2]

EnYear is another company that prints and sells T-shirts. The price, $N$ euros, that they charge for an order can be modelled by the linear function $N(x) = 9x$, where $x$ is the number of T-shirts in the order.

(c) Write down the number of T-shirts in an order for which EnYear charged 63 euros.[1]

An order of $p$ T-shirts will be charged the same price by both M-Line and EnYear.

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

The total cost, $C(d)$, in Canadian dollars (CAD), to hire a bicycle for $d$ days from *Pedal Paradise* is given by the function

$$C(d) = 60d + 10, d \geq 3, d \in \mathbb{Z}$$

The total cost includes a fixed charge to hire both a helmet and a repair kit.

(a) State, in context, what the values 60 and 10 represent.[2]

(b) Calculate the cost of hiring a bicycle for 5 days.[2]

Hema hires a bicycle from *Pedal Paradise*.

(c) Write down the minimum number of days she can hire the bicycle.[1]

(d) Given that $C^{-1}(1270) = k$, find the value of $k$.[2]

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]

A quadratic function $f$ is given by $f(x) = ax^2 + bx + c$. The points $(0, 5)$ and $(-4, 5)$ lie on the graph of $y = f(x)$.

(a) Find the equation of the axis of symmetry of the graph of $y = f(x)$.[2]

(b) Write down the value of $c$.[1]

The $y$-coordinate of the minimum of the graph is 3.

(c) Find the value of $a$ and of $b$.[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 $R(t) = 100e^{-pt}$, $t \geq 0$, 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]

"Password entropy" is a measure of the predictability of a computer password. The higher the entropy, the more difficult it is to guess the password.

The relationship between the password entropy, $p$ (measured in bits) and the number of guesses, $G$, required to decode the password is given by $0.301p = \log_{10} G$.

(a) Calculate the value of $p$ for a password that takes 5000 guesses to decode.[2]

(b) Write down $G$ as a function of $p$.[1]

(c) Find the number of guesses required to decode a password that has an entropy of 28 bits. Write your answer in the form $a \times 10^k$, where $1 \leq a < 10, k \in \mathbb{Z}$.[3]

There is a point on the graph of the function $G(p)$ with coordinates $(0, 1)$.

(d) Explain what these coordinate values mean in the context of computer passwords.[1]

Elvis Presley is an extremely popular singer. Although he passed away in 1977, many of his fans continue to pay tribute by dressing like Elvis and singing his songs.

The number of Elvis impersonators, $N(t)$, can be modelled by the function

$$N(t) = 170 \times 1.31^t,$$

where $t$ is the number of years since 1977.

(a) Write down the number of Elvis impersonators in 1977.[1]

(b) Calculate the time taken for the number of Elvis impersonators to reach 130000.[2]

(c) Calculate the number of Elvis impersonators when $t = 70$.[2]

(d) Use this information to explain why the model for the number of Elvis impersonators is unrealistic.[1]

The world population in 2047 is projected to be 9500000000 people.

Consider the graph of the function $f(x) = \frac{3}{x} - 2, x \neq 0$.

(a) Write down the equation of the vertical asymptote.[2]

(b) Write down the equation of the horizontal asymptote.[2]

(c) Calculate the value of $x$ for which $f(x) = 0$.[2]

The intensity level of sound, $L$ measured in decibels (dB), is a function of the sound intensity, $S$ watts per square metre ($\mathrm{W m}^{-2}$). The intensity level is given by the following formula.

$$L = 10\log_{10}(S \times 10^{12}), S \geq 0$$

(a) An orchestra has a sound intensity of $6.4 \times 10^{-3} \mathrm{W m}^{-2}$. Calculate the intensity level, $L$ of the orchestra.[2]

(b) A rock concert has an intensity level of 112 dB. Find the sound intensity, $S$.[2]