Functions, domain, range: notes and practice questions

(a) A scientist is observing the trajectory of a particle. The relationship between its horizontal position ($x$) and vertical position ($y$) is described by the equation $x^2 + y^2 = 9$.

Does this relationship represent $y$ as a function of $x$? Justify your answer.

(b) Another experiment records the temperature ($y$) of a chemical reaction over time ($x$). The relationship is modelled by the equation $y = x^2 - 4x + 3$.

Does this relationship represent $y$ as a function of $x$? Justify your answer.

(c) A sensor measures a constant environmental factor, where its value ($x$) is always $2$, regardless of any other variable ($y$). The relationship is given by $x = 2$.

Does this relationship represent $y$ as a function of $x$? Justify your answer.

(d) A technician plots the oscillation of a spring, where its displacement ($y$) is related to time ($x$) by the equation $y = \sin(x)$.

Does this relationship represent $y$ as a function of $x$? Justify your answer.

In a controlled chemical experiment, the initial temperature, $x$ (in degrees Celsius), of a reactant mixture is adjusted before a reaction begins. The adjusted temperature, $f(x)$, is then used to determine the reaction rate, $g(f(x) )$.

The adjustment function is given by $f(x) = x - 3$, for $x \in \mathbb{R}$.

State the range of $f(x)$.

The overall reaction rate, $gf(x)$, as a function of the initial temperature $x$, is modelled by $gf(x) = 4x^2 - 24x + 30$, for $x \in \mathbb{R}$.

State the range of $gf(x)$.

The experiment is designed to achieve a specific 'target state' when the adjusted reaction rate $f(gf(x) )$ equals 0. Solve the equation $fgf(x) = 0$.

Determine the function $g(x)$.

(a) Consider the graph of a relationship where the input is the length of a side of a square, $x$, and the output is its area, $A$. This graph is represented by $A = x^2$ for $x \ge 0$. State whether this graph represents $A$ as a function of $x$, giving a reason.

(b) Consider the graph of a relationship between two variables, $x$ and $y$, which forms a vertical line segment from $(3, 1)$ to $(3, 5)$. State whether this graph represents $y$ as a function of $x$, giving a reason.

The "LearnFast" online learning platform charges a monthly subscription fee of $\$40$ and an additional $\$7$ per premium course. If a student enrolls in a minimum of $10$ premium courses in a month, a one-time discount of $\$25$ is applied to their total bill.

This can be modelled by the following function, $L$, which gives the total cost when enrolling in a minimum of $10$ premium courses at LearnFast:

$L(x) = 7x + 15, x \ge 10$

where $x$ is the number of premium courses a student enrolls in.

Find the total cost of enrolling in $20$ premium courses at LearnFast.

Find $L^{-1}(99)$.

Another online learning platform, "SkillUp", charges a flat rate of $\$8$ per premium course, with no monthly subscription or discounts. A student must enroll in a minimum of $10$ premium courses for a direct comparison with LearnFast's discounted model.

The total cost at LearnFast is cheaper than SkillUp when $x > k$.

Find the minimum integer value of $k$.

(a) An engineer is designing a section of a roller coaster track. To ensure a smooth ride, the track must follow a specific curve. Three sensor readings are taken at different horizontal positions along this section of the track, giving the following height coordinates: $(1, 4)$, $(2, 7)$, and $(3, 14)$.

Assuming the track follows a quadratic path of the form $y = ax^2 + bx + c$, find the equation of this quadratic curve.

(b) A fourth sensor reading is taken at a horizontal position of $x=4$, recording a height of $y=28$. Determine if this new point lies on the quadratic curve found in part (a).

(c) Due to the discrepancy, the engineer decides that a cubic function, $y = Ax^3 + Bx^2 + Cx + D$, would be a better fit for the track. Find the equation of the cubic function that passes through all four points: $(1, 4)$, $(2, 7)$, $(3, 14)$, and $(4, 28)$.

(a) The height of a drone above the ground, $h$ metres, $t$ seconds after launch, is modelled by a function whose graph is shown below. The drone is launched from a platform, reaches a maximum height, and then descends to the ground. The graph starts at $(0, 7)$, reaches a maximum point at $(3, 16)$, and touches the horizontal axis at $(7, 0)$.

State the domain and range for this function.

(b) A freshly brewed cup of coffee is left to cool. Its temperature, $T$ degrees Celsius, $t$ minutes after brewing, is modelled by a function whose graph is shown below. The graph starts at $(0, 100)$, and as time progresses, the temperature decreases, approaching an ambient temperature of $25$ degrees Celsius, but never quite reaching it.

State the domain and range for this function.

A manufacturing company purchases a new specialized machine. The machine's value depreciates exponentially over time. The value of the machine, $V$, in thousands of dollars, $t$ years after its purchase, is modelled by the function $V(t) = A e^{-kt}$, for $t \ge 0$.

The initial value of the machine was 150 thousand dollars. After 3 years, its value had decreased by 40%.

(a) Find the value of $k$.

(b) Calculate the value of the machine after 5 years and 6 months, giving your answer to two decimal places.

(c) The company believes that, according to this model, the machine will always have some residual value, however small.

State a mathematical reason why the company might believe this.

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

The concentration of a certain chemical, $C$, in a solution over a period of time can be modelled using the function $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$, where $t$ is the time in hours after the experiment begins.

Sketch the graph of $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$ for $0 \le t \le 20$.

Find the concentration after $2$ hours.

Find the concentration after $15$ hours.

Find the maximum concentration and the time in hours at which this occurs.

Find the minimum concentration and the time in hours at which this occurs.

Find the times in hours when the concentration is $6$ mol/L.

An architect is designing a parabolic arch for a new building. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $x$ and $f(x)$ are measured in metres. The highest point of the arch (the vertex) is at $(2, 4)$. One end of the arch is at the origin $(0, 0)$, and the other end is at $(p, 0)$.

(a) Find the value of $p$.

(b) Find the value of

(i) $a$.

(ii) $b$.

(iii) $c$.

(c) Write down the equation of the axis of symmetry of the arch.

A drone launches a package, and its trajectory is modelled by the equation $h(x) = -0.015x^2 + 0.6x + 5$, where $h(x)$ is the height of the package in metres and $x$ is the horizontal distance in metres from the launch point.

On paper, sketch the graph of the path that the package flies for $x \ge 0$. Clearly indicate the initial height, the maximum height, and the horizontal distance when it lands.

Find the height of the package when it has travelled a horizontal distance of $15$ metres.

Find the maximum height of the package.

Find the horizontal distance at which the package lands on the ground, and explain what this value represents in the context of the problem.

A new fertilizer is being tested on a specific plant species. The concentration of a key nutrient, $C$, in the plant (in mg/L) is modelled by the function $C(x) = 5 - \frac{30}{x-2}$, where $x$ is the amount of fertilizer added (in grams). The amount of fertilizer used is restricted to $-4 \le x \le 8$, where $x \neq 2$ (due to experimental constraints).

Find the range of the nutrient concentration $C(x)$.

Determine the amount of fertilizer $x$ that results in a nutrient concentration of $-1$ mg/L. Give your answer in the form $C^{-1}(-1)$.

(a) A pharmaceutical company is designing a new cylindrical container for a special liquid. The container has a fixed volume $V = 500 \text{ cm}^3$.

The radius of the container is $r \text{ cm}$ and the height is $h \text{ cm}$.

Show that $\pi r^2 h = 500$.

(b) Find an expression for the total surface area $S$ of the container.

(c) Substitute an expression for $h$ (from part (a) ) into your expression for $S$ (from part (b) ) and hence show that $S = 2\pi r^2 + \frac{1000}{r}$.

(d) Find $\frac{dS}{dr}$.

(e) Find the minimum value of $S$ and the values of $r$ and $h$ when this occurs. Show that this value of $S$ is indeed a minimum.

Let the function $C(w)$ represent the cost in dollars per display stand, where $w$ is the width of the stand in meters.

$C(w) = \frac{500}{w^2} + 1.2$ for $5 \le w \le 15$.

Find the range of $C$.

The function $C^{-1}$ is the inverse function of $C$.

Find $C^{-1}(15)$.

In the context of the question, interpret your answer to part (b)(i).

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

The "SkyGazer" is a new observation wheel in a city park. The wheel has a diameter of $70$ m. To begin the ride, a passenger enters a capsule at the lowest point on the wheel, which is $3$ m above the ground. A ride consists of multiple revolutions, and the wheel makes $2$ revolutions per minute.

The height of a capsule above the ground, $h$, measured in metres, during a ride on the SkyGazer can be modelled by the function $h(t) = -a \cos (bt) + d$, where $t$ is the time, in seconds, since a passenger began their ride.

(a) Calculate the value of

(i) $a$;

(a)(ii) $b$;

(a)(iii) $d$.

(b) A ride on the SkyGazer lasts for $10$ minutes in total.

Calculate the number of revolutions of the wheel per ride.

(c) For exactly one ride on the SkyGazer, suggest

(i) an appropriate domain for $h(t)$;

(c)(ii) an appropriate range for $h(t)$.

(d) A $20$ metre-tall building stands on the horizontal ground next to the SkyGazer.

By considering the graph of $h(t)$, determine the length of time during one revolution of the wheel for which the capsule is higher than the building.

(e) There is a plan to relocate the SkyGazer onto a taller platform which will increase the maximum height of the wheel to $75$ m. This will change the value of one parameter, $a$, $b$ or $d$, found in part (a).

(i) Identify which parameter will change.

(e)(ii) Find the new value of the parameter identified in part (e)(i).

A biologist is studying the effect of a new toxin on a bacterial culture. The number of active bacterial cells, $N$ (in thousands), remaining after exposure to a toxin dose $D$ (in mg/L) follows the relationship:

$\log_{10}N = k - D$, for some constant $k \in \mathbb{R}$.

In an experiment, it was observed that when the toxin dose was 2 mg/L, there were 1000 thousand (i.e., 1,000,000) active cells remaining.

(a) Find the value of $k$.

The relationship for this bacterial culture can also be written in the form $N = \frac{b}{10^D}$.

(b) Find the value of $b$.

(c) Given that the toxin dose $D$ is between 0.5 mg/L and 4.0 mg/L (i.e., $0.5 < D < 4.0$ ), find the range for $N$.

The effectiveness score, $S$, of a new antidote is inversely proportional to the number of active cells, $N$, remaining after toxin exposure, such that $S = \frac{1}{N}$.

(d) A new antidote was tested on a culture exposed to a dose of 4.5 mg/L. Calculate the effectiveness score, $S$, for this antidote. Give your answer to 3 significant figures.

A tech company launches a new social media app. The number of active users, $U$, can be modelled by the function

$U(t) = 800 \times k^t, t\ge 0$,

where $t$ is the number of hours since the app was launched, and $k$ is a positive constant.

Write down the value of $U(0)$.

Interpret what this value means in this context.

4 hours after the app was launched, the number of active users was 4050.

Find the value of $k$.

Find the number of active users 2 hours and 15 minutes after the app was launched.

A competitor launches a similar app, whose user base, $U_2$, can be modelled by the function

$U_2(t) = 2500 \times 1.2^t, t\ge 0$,

where $t$ is the number of hours since both apps were launched.

Find the value of $t$ when the number of users for both apps is equal.

It takes $H$ hours and $m$ minutes for the number of users of the first app to reach 15000.

Find the value of $H$ and $m$, giving $m$ as an integer.

Each user of the first app requires $1.5 \times 10^{-2}$ MB of server storage. The total available server capacity is $3.0 \times 10^5$ MB.

Determine how long it would take for the app's user base to exceed the server capacity.

A new experimental drug is administered to a patient. The concentration of the drug in the patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modelled by the function $C(t) = C_0 e^{-kt}$, where $C_0$ and $k$ are positive constants.

Initially, the concentration of the drug is $250$ mg/L. After $2$ hours, the concentration drops to $150$ mg/L.

Determine the value of $k$.

Using this model, calculate the concentration of the drug in the bloodstream $5$ hours after administration.

Based on this model, will the drug ever completely leave the patient's bloodstream? Justify your answer.

State one limitation of the domain of this model in a real-world context.

(a) A confectionery company is designing a new wrapper for a chocolate bar. The wrapper is in the shape of a rectangular prism with a square base of side length $x$ cm. The height, $h$ cm, is twice the side length of the base.

Write down an expression for $h$ in terms of $x$.

(b) The chocolate bar has a volume of $250$ cm$^3$.

Find the value of $x$ and $h$.

(c) Calculate the total external surface area of this rectangular prism wrapper.

(d) The company also considers a cylindrical wrapper with radius $r$ cm and height $H$ cm. This wrapper must also hold $250$ cm$^3$ of chocolate.

Find an expression for the height, $H$, of the cylindrical wrapper in terms of $r$.

(e) Let the total external surface area of the cylindrical wrapper be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{dA}{dr}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ for the cylindrical wrapper.

(h) To account for manufacturing waste and overlap, an additional $12\%$ of the calculated surface area is required for the rectangular prism wrapper, and $20\%$ for the cylindrical wrapper.

Determine which wrapper design the company should choose to minimize material usage. Justify your answer.

A scientist is modeling the concentration of a certain chemical in a solution over time. The concentration, $C$, in mg/L, is given by the function $C(t) = 4 + \frac{18}{t-3}$, where $t$ is the time in hours. The model is valid for $1 \leq t \leq 9$, but the chemical reaction causes a singularity at $t=3$ hours, so $t \neq 3$.

(a) Find the range of $C$.

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

(c) Write down the range of $C^{-1}(t)$.