Composite & inverse functions: notes and practice questions

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

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 $g(x)$.

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 $C^{-1}$.

(a) A manufacturing company uses a specialized machine to produce custom parts. The number of units produced, $N$, in one day is modelled by

$N = 40 \ln(3t+1)$, for $0 \le t \le 10$

where $t$ is the time the machine operates, in hours, on that day.

Find the time, in hours, it takes for the machine to produce 80 units in one day.

(b) The daily profit, $P$, in thousands of dollars, from producing $N$ units is given by

$P = 150(1-e^{-0.008N})$, for $P \in \mathbb{R}$.

Find the profit, in thousands of dollars, the company earns for producing 80 units.

(c) Find an expression for $P$ as a function of $t$, giving your answer in the form

$P=150(1-\frac{1}{(3t+1)^n})$

where $n$ is a number to be determined.

(d) Hence or otherwise, find the profit, in thousands of dollars, the company earns by operating the machine for 4 hours.

(e) Find the greatest profit, in thousands of dollars, the company can earn in one day.

(f) The company introduces a weekly bonus profit, $B$, in thousands of dollars, for the total units produced in one week, $D_{total}$, using

$B=100(1-\frac{1}{D_{total}+1})$

The company decides to operate the machine for the same length of time, $T$ hours, each day for 5 days a week.

They want to achieve a total weekly profit of 580 thousand dollars.

Find the value of $T$.

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 $C^{-1}(t)$.

Two geometric transformations, H and K, are defined on points $(x, y)$ in a Cartesian plane. The transformations are given by the functions $h, k: ℝ^2 → ℝ^2$ defined by:

$h((x, y)) = (x - 2y, 2x + y)$

$k((x, y)) = (y, -x)$

(a) (i) Find the composite transformation $(h ∘ k)((x, y))$.

(ii) Find the composite transformation $(k ∘ h)((x, y))$.

(b) The order in which transformations are applied can be important. Determine, with a reason, whether the transformations H and K are commutative.

(c) Find the inverse transformation $h⁻¹$.

A company produces custom-sized square display screens. The production cost, $C(s)$, in dollars, for a screen with side length $s$ cm is modelled by the function:

$C(s) = \frac{1500}{s} + 25$, for $15 \le s \le 75$.

Find the range of $C(s)$.

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

Find $C^{-1}(70)$.

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

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

A new automated packaging machine's efficiency, $E$, is modelled by a function of the number of hours, $t$, it has been operating since its last maintenance. The efficiency score is given by $E(t) = \frac{3}{\sqrt{t+1}} + 2$, where $t$ is in hours and $t > 0$.

(a) Calculate the number of hours, $t$, for which the machine's efficiency score is 3.

(b) Determine the domain of the inverse function $E^{-1}(x)$.

A biologist is studying the population growth of a certain species of algae in a pond. The population, $P$, in thousands of cells per millilitre, $t$ days after the start of the study, is modelled by the function $P(t) = 2e^t - 3$, for $t \ge 0$.

Find an expression for the inverse function, $P^{-1}(t)$. You are not required to state a domain.

The biologist is interested in the specific time(s) when the numerical value of the population (in thousands of cells per millilitre) is equal to the numerical value of the time (in days).

Solve $P(t) = P^{-1}(t)$.

A chemical reaction's rate, $R$, in mol/L per minute, depends on the time, $t$, in minutes since the reaction started. The relationship is modelled by the function:

$R(t) = \frac{6}{\sqrt{t+3}} - 1, \text{ for } t \ge 0$

Calculate the time (in minutes) when the reaction rate is $2 \text{ mol/L per minute}$.

Determine the range of possible reaction rates for this model, and hence state the domain of $R^{-1}(x)$.

(a) The formula for converting daily steps, $S$, to a fitness score, $F$, is given by $F = 0.05S + 10$.

(i) Find a formula for converting a fitness score, $F$, back to daily steps, $S$.

(ii) A user achieved a fitness score of 75. Calculate the number of daily steps they took.

(b) Over a month, the mean daily steps recorded by a group of users was 8500 steps with a standard deviation of 1200 steps.

For the same group, find

(i) the mean daily fitness score.

(ii) the standard deviation of the daily fitness scores.

A patient is administered a dose of a certain medication. The concentration of the drug in their bloodstream, $C$, measured in mg/L, $t$ hours after administration, can be modelled by the function:

$C(t) = 120 e^{-0.15t} + 5$, $t \ge 0$

Find the concentration of the drug in the bloodstream 8 hours after administration.

State the long-term concentration of the drug in the bloodstream.

The drug is considered effective as long as its concentration is at least 30 mg/L. Determine the time, in hours, when the drug concentration first drops to 30 mg/L.

The total cost, $C(m)$, in US dollars (USD), to subscribe to 'StreamVerse Premium' for $m$ months is given by the function $C(m) = 18m + 25$, where $m \ge 2$ and $m \in \mathbb{Z}$.

The total cost includes a one-time activation fee.

State, in context, what the values 18 and 25 represent.

Calculate the cost of subscribing to StreamVerse Premium for 7 months.

A user wants to subscribe to StreamVerse Premium.

Write down the minimum number of months they can subscribe for.

Given that $C^{-1}(385) = k$, find the value of $k$.

The graph of the function $f$ is shown in the following diagram, representing the concentration of a certain chemical in a solution over time $x$ (in minutes).

(a) Write down the concentration of the chemical at time $x=1$ minute.

(b) On the same axes, sketch the graph of $y = f^{-1}(x)$, representing the time required to reach a certain concentration.

A different chemical's concentration, $h(x)$, is modeled by the function $h(x) = 2x - 3$, where $x$ is time in minutes.

(c) Find an expression for $h^{-1}(x)$.

(d) Find a value of $x$ where $f^{-1}(x) = h^{-1}(x)$.

A company's monthly revenue, $R$ (in thousands of dollars), is modelled by a linear function of the number of units, $x$ (in hundreds), sold. The function is given by $R(x) = ax + b$.

When 200 units are sold, the revenue is $1000. When 500 units are sold, the revenue is$7000.

(a) Determine the value of $a$ and the value of $b$.

(b) Find an expression for $R^{-1}(x)$.

(c) Solve the equation $R(x) = R^{-1}(x)$, giving your answer in an exact form.

(d) Explain why, for any function $h(x)$, the equation $h(x) = h^{-1}(x)$ will have the same solution(s) as the equation $h(x) = x$.

The two graphs illustrate relationships within a specialized coffee roasting company.

Write down which graph represents direct variation, with a reason.

(b) Find and interpret the gradient of each graph in context.

(c) Find an equation for the roasting time $T$ (in minutes) as a function of the batch size, $S$ (in number of bags), and use it to predict how long the roasting process will be for a batch of $1500$ bags.

A local delivery service charges for deliveries based on the distance, $x$ km, from the origin. The cost, $C(x)$ in dollars, is modelled by the piecewise function:

$C(x) = \begin{cases} 1.5x + 5 & 0 \le x \le 10 \\ 2x & x > 10 \end{cases}$

(a) Find the inverse function, $C^{-1}(x)$, and state its domain and range.

A different data plan charges users based on the amount of data, $x$ GB, consumed. The cost, $P(x)$ in dollars, is given by:

$P(x) = \begin{cases} 5x + 10 & 0 \le x \le 5 \\ -2x + 45 & x > 5 \end{cases}$

(b) Show that the function $P(x)$ is not invertible.

A company's daily profit, $P(n)$ (in thousands of dollars), depends on the number of units, $n$ (in hundreds), it sells. The profit function is given by $P(n) = 4n - 5$.

On a particular day, the company's profit was between $23$ thousand dollars and $75$ thousand dollars (exclusive). Find the possible range for the number of units sold, $n$.

Determine the inverse function, $P^{-1}(n)$.

State the range of the inverse function $P^{-1}(n)$.

A function $f$ is defined by $f(x) = e^x + 1$. Let $g$ be the inverse function of $f$.

(a) (i) State the $y$-intercept of the graph of $y = f(x)$.

(ii) State the coordinates of the point where the graph of $y = g(x)$ cuts the $x$-axis.

(b) Determine an equation for $g(x)$ in the form $y = g(x)$.

(c) (i) Write down the equation of the horizontal asymptote to the graph of $y = f(x)$.

(ii) Write down the equation of the vertical asymptote to the graph of $y = g(x)$.

(d) Hence state the domain of $g$.

(e) Solve the equation $f(x) + g(x) = 0$.

(a) A travel agency, GlobalVoyages, offers various package deals. For flight packages priced over $1000, they apply a 15% discount.

Another travel agency, WorldExplorer, offers a fixed reduction of $200 on all hotel packages priced over$1000.

Given that $x$ is the original price of a travel package and $x > 1000$,

(i) write down a function $f(x)$ that models the final price of a flight package at GlobalVoyages, after the discount is applied.

(ii) write down a function $g(x)$ that models the final price of a hotel package at WorldExplorer, after the reduction is applied.

(b) A third agency, GrandTour, decides to offer both a 15% discount and a $200 reduction on all their travel packages priced over$1000.

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

(c) State, with justification, whether $(f \circ g)(x)$ or $(g \circ f)(x)$ is better for the customer when they buy a travel package with an original price of $1800.