Exercises: Inverse proportion functions: 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 that relates 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 are located when graphed:

1. a) $y = \frac{5}{x}$
2. b) $y = \frac{-2}{x}$
3. c) $y = \frac{0,1}{x}$

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

To travel a fixed distance of $120$ miles, the time $t$ taken (in hours) and the average speed $v$ (in mph) 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 trip.

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 the rational function $y = \frac{7}{x+5}$.

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

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

According to Boyle's law, at a constant temperature, the pressure $P$ of a gas and its volume $V$ are inversely proportional quantities ($P \cdot V = k$). 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.

The graph of the function $y = \frac{8}{x}$ is translated in the Cartesian plane $4$ units to the left and $3$ units down. 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 points of intersection with the coordinate axes of the rational function $y = \frac{10}{x-2} + 5$.

A group of people rents a tennis court for $\$24$. Write the function $C(x)$ that expresses the cost per person as a function of the number of players $x$. If the final cost was $\$6$ per person, how many players rented the court? What is the logical domain of this function taking into account the context of the problem?

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 $(0, 5)$.

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 $R$ as a function of $S$. What must the minimum cross-section of the wire be so that the resistance is less than or equal to $0,01\ \Omega$?

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 in the form $y = \frac{k}{x-a} + b$.

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 $y = -x + 4$.

An electronic components factory has fixed daily expenses of $\$500$ to turn on the machinery, and producing each unit costs $\$2$. Write the function for the average cost per unit $C_m(x)$ as a function 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.

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 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 the point of intersection with the y-axis.

The focal length of a thin converging lens is $20$ inches. 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 $s_i$ as a function of $s_o$. What happens to the image distance when the object approaches $20$ inches from the lens from the right?

Determine the domain, the range, the asymptotes and the points of intersection with the axes of the function $y = \frac{-3}{x+2} - 4$. Also state its intervals of increase and decrease.