Inverse Proportionality Functions

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 $y = \frac{k}{x}$.

State the equations of the vertical and horizontal asymptotes of the hyperbola whose algebraic expression is $y = \frac{5}{x-4} + 3$.

Find the domain of definition of the rational function $y = \frac{7}{x+5}$.

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.

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 $(0, 5)$.

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

A hyperbola of the form $y = \frac{k}{x-a} + b$ has its center of symmetry at the point $(-1, 3)$ and intersects the x-axis at $x = 1$. Find its algebraic expression and calculate its y-axis intercept.

Determine the domain, the range, the asymptotes and the axes intercepts of the function $y = \frac{-3}{x+2} - 4$. State also its intervals of increase and decrease.

Express the rational function $y = \frac{2x+7}{x+3}$ in the form $y = \frac{k}{x-a} + b$ by performing polynomial division. Then, state the equations of its asymptotes and the coordinates of its center of symmetry.

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

Find the value of the constant $k$ so that the hyperbola $y = \frac{k}{x}$ is tangent to the line $y = -x + 6$ (that is, so that they intersect at a single point).

A lottery prize of $6\,000$ pesos 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 cover a fixed distance of $120$ km, the time $t$ taken (in hours) and the average speed $v$ (in km/h) are inversely proportional quantities. Write the function $v(t)$ and calculate at what constant speed a car must travel if it wants to take exactly $2.5$ hours to complete the journey.

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 $20$ liters at a pressure of $1.5$ atmospheres, write the function $V(P)$ and calculate what volume it will occupy at $3$ atmospheres.

A group of people hires a padel court for $480$ pesos. Write the function $C(x)$ expressing the cost per person in terms of the number of players $x$. If the final cost was $120$ pesos 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 $10\,000$ pesos to turn on the machinery, and producing each unit costs $40$ pesos. 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.

The focal length of a thin converging lens is $20$ cm. The relationship between the object distance to the lens $s_o$ and the image distance $s_i$ is given by the equation $\frac{1}{20} = \frac{1}{s_o} + \frac{1}{s_i}$. Express $s_i$ in terms of $s_o$. What happens to the image distance when the object approaches $20$ cm from the lens from the right?