Inverse proportion: y = k/x

Identify in the following table whether the quantities $x$ and $y$ are inversely proportional and, if so, find the constant of proportionality $k$ and write the function relating them.

• $x$: 1, 2, 3, 4, 6

• $y$: 24, 12, 8, 6, 4

Given the inverse proportion function $y = \frac{18}{x}$, calculate the value of $y$ for $x = 3$, $x = -9$ and $x = 0.5$.

Determine the domain, the range and the equations of the asymptotes of the function $y = \frac{-8}{x}$.

Match each of the following functions with the description of the quadrants in which their branches lie when plotted graphically:

1. a) $y = \frac{5}{x}$
2. b) $y = \frac{-2}{x}$
3. c) $y = \frac{0.1}{x}$

Descriptions:

1. Branches located in the first and third quadrants.

2. Branches located in the second and fourth quadrants.

Look at the following graph of an inverse proportion function and determine its constant $k$ and its algebraic expression $y = \frac{k}{x}$.

A lottery prize of $300$ € is to be divided equally among a group of $x$ friends. Write the algebraic expression of the function that relates the money each person receives to the number of friends, and calculate how much they receive if there are $6$ people.

To fill a swimming pool, $4$ taps with equal flow rate take $9$ hours. Write the function relating the filling time $t$ (in hours) to the number of open taps $n$. How long would $6$ taps take to fill the same pool?

According to Boyle's law, at constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \times V = k$). If a given gas occupies a volume of $20$ litres at a pressure of $1.5$ atmospheres, write the function $V(P)$ and calculate what volume it will occupy at $3$ atmospheres.

The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units downwards. Write the algebraic expression of the new function, determine its domain and find the coordinates of the point where its asymptotes intersect.

Aoife is studying the monotonicity of the function $f(x) = \frac{8}{x}$. She knows that the constant is $k = 8 > 0$ and her teacher explained in class that functions with $k > 0$ are decreasing. To check this empirically, she takes two values of the variable: $x_1 = -2$ and $x_2 = 4$. Calculating their images, she gets $f(-2) = -4$ and $f(4) = 2$. Surprised, she observes that $-2 < 4$ and yet $f(-2) < f(4)$, so she concludes that “the theory fails, because moving to the right the function has gone up”. Identify and explain precisely Aoife's conceptual error, clarify why her example does not contradict the theory, and state how the decrease of $f(x)$ must be phrased in a mathematically rigorous way.

From the graph of this translated hyperbola, determine the equations of its asymptotes, the coordinates of its centre of symmetry and deduce its algebraic expression of the form $y = \frac{k}{x-a} + b$.

Find analytically the axes intercepts of the rational function $y = \frac{10}{x-2} + 5$.

Write the equation of a hyperbola of the form $y = \frac{k}{x-a} + b$ knowing that its asymptotes are the lines $x = -2$ and $y = 4$, and that its graph passes through the point $(0, 5)$.

Find analytically the points of intersection between the hyperbola $y = \frac{-4}{x-1}$ and the line $y = -x + 4$.

A group of people hires a padel court for $24$ €. Write the function $C(x)$ expressing the cost per person in terms of the number of players $x$. If the final cost was $6$ € per person, how many players hired the court? What is the logical domain of this function taking into account the context of the problem?

The electrical resistance $R$ (in ohms, $\Omega$) of a conducting wire of constant length is inversely proportional to its cross-sectional area $S$ (in mm$^2$). If for a cross-section of $2 \text{ mm}^2$ the resistance is $0.02\ \Omega$, express $R$ in terms of $S$. What must the minimum cross-section of the wire be so that the resistance is less than or equal to $0.01\ \Omega$?

An electronic components factory has daily fixed overheads of $500$ € to turn on the machinery, and producing each unit costs $2$ €. Write the average cost function per unit $C_m(x)$ in terms of the number of units $x$ manufactured per day. Find its asymptotes and interpret the meaning of the horizontal asymptote in the context of the problem.