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 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.

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 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.

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

Lucía 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 Lucía'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.

Find the algebraic expression of a translated inverse proportionality function, of the form $g(x) = \frac{k}{x-h} + v$, knowing that it simultaneously satisfies these two conditions: its center of symmetry is the point $(-2, 3)$ and its graph passes through the point $(1, 2)$. Using the obtained function:

1. a) Determine the numerical value of the constants $k$, $h$, and $v$.
2. b) Write the equations of its vertical and horizontal asymptotes, and state its domain and range, justifying which value is excluded in each case.
3. c) Reason, from the sign of the constant $k$, whether the branches of the graph are increasing or decreasing on each interval of its domain.

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, $4$ 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 ($P \cdot V = k$). 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.

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

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?

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

A group of 20 4th-year ESO students from Zaragoza organizes an end-of-year trip and hires a bus for a flat rate of $1\,200\text{ \euro}$, to be split equally among those traveling. Before finalizing the booking, the school management allows opening registration to more students.

1. a) Before doing any exact calculation: if the number of attendees increases from 20 to 30 students (a $50\text{ \%}$ increase), will the individual cost be reduced to half, to more than half, or to less than half of the original price? Make a qualitative estimate and justify your answer based on the relationship between the quantities.
2. b) Model the function $P(x) = \frac{k}{x}$ expressing the price per student as a function of the total number of attendees $x$. Calculate the individual amount for $x = 20$ and for $x = 30$, calculate the actual percentage discount experienced by each ticket, and check whether your initial estimate was correct.
3. c) If the bus has a maximum capacity of 55 seats, what is the minimum possible price per person? Could it ever reach $0\text{ \euro}$ if infinitely many students were to travel? Interpret this limit using the horizontal asymptote of the function.

In a $30\text{ km}$ roadwork section on a dual carriageway in Castilla y León, the DGT sets temporary restrictions. The time $t$ (in hours) needed to travel along this section depends on the average speed $v$ (in $\text{km/h}$) at which a vehicle travels according to the relation $t(v) = \frac{30}{v}$.

1. a) Justify why $t$ and $v$ are inversely proportional quantities and state the units and the physical meaning of the constant $k = 30$.
2. b) Analyze the asymptotes of the function in the context of this journey: what physical meaning would approaching the vertical asymptote $v = 0$ have? And the horizontal asymptote $t = 0$? Reason whether it is physically possible to reach either of the two.
3. c) A driver argues: “If I reduce my speed from $120\text{ km/h}$ to $60\text{ km/h}$, the time I lose is exactly the same as if I reduce it from $60\text{ km/h}$ to $30\text{ km/h}$, because in both cases I halved the speed”. Prove quantitatively whether their statement is true or false by calculating the minutes of delay in each case, and explain what property of the hyperbola causes the time losses not to be equal.

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

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 centre 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 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.

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 centre 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.

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?

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.

You have been asked to plot and analyze in detail the homographic function $f(x) = \frac{3x - 1}{x - 2}$, identifying its asymptotes, its center of symmetry, and its monotonicity. Two classmates suggest different methods:

• Method 1 (Marcos): Build a table with 10 very large values of $x$ (positive and negative) and several very close to $x = 2$, to graphically guess where the lines approached by the curve pass.

• Method 2 (Sara): Carry out polynomial division between numerator and denominator to express the function in the canonical form $f(x) = \frac{k}{x-h} + v$, and directly deduce its properties from geometric transformations.

1. a) Carry out Method 2: perform the algebraic division of $(3x - 1)$ by $(x - 2)$ and write $f(x)$ in the form $\frac{k}{x-h} + v$.
2. b) From the canonical expression you obtained, state with justification the equations of the two asymptotes, the exact coordinates of the center of symmetry, and deduce whether the function is increasing or decreasing on each of its two branches.
3. c) Critically evaluate both methods: explain why Sara's algebraic method is superior to Marcos's numerical method for avoiding approximation errors and deducing asymptotic behavior with analytical certainty.

It is desired to analytically reconstruct a translated hyperbola $g(x) = \frac{k}{x - h} + v$ from a series of geometric and analytical conditions:

1. Its graph intersects the x-axis at the point $(6, 0)$.

2. The center of symmetry of the curve is the point $(4, -2)$.

3. Its graph intersects the y-axis at the point $(0, -3)$.

1. a) Reasonably determine the numerical values of the parameters $h$, $v$, and the constant $k$.
2. b) Algebraically check condition 1: set up the equation $g(x) = 0$ with the values obtained and solve it to verify that its solution is $x = 6$.
3. c) Given that the center of symmetry is $(h, v)$, calculate the coordinates of the point on the curve that is symmetrical with respect to this center to the y-intercept, $(0, -3)$, and verify analytically that this symmetrical point satisfies the equation of $g(x)$.