Exercises: Inverse proportion function

A student calculates the asymptotes of the function $f(x) = \frac{4x+1}{x-2}$ and concludes that it has a horizontal asymptote at $y=0$ because "all rational functions always tend to zero at infinity". Identify the error in their rule of thumb and find the correct equation of the horizontal asymptote.

Given the following table relating the base $x$ and the height $y$ of different rectangles with the same area:

• $x$: $2$, $3$, $4$, $6$

• $y$: $12$, $8$, $6$, $4$

Determine whether the relationship between $x$ and $y$ corresponds to an inverse proportion function. If so, find the constant of proportionality $k$ and write the algebraic expression for the function.

Consider the inverse proportionality function $f(x) = \frac{-12}{x}$, whose graphical representation is shown below:

Based on its expression and graph, answer the following questions:

1. a) State its domain and range.
2. b) Write the equations of its asymptotes.
3. c) Determine whether the function is increasing or decreasing on its domain.

Find the algebraic expression of the inverse proportion function of the form $y = \frac{k}{x}$, knowing that its graph passes through the point $P(-2, 5)$.

Given the function $g(x) = \frac{3}{x+4} - 2$, state the equations of its horizontal and vertical asymptotes, and the coordinates of the centre of symmetry of the hyperbola.

A train travels a fixed distance of $120\text{ km}$. Write the function relating the time $t$ (in hours) it takes to complete the journey as a function of the average speed $v$ (in $\text{km/h}$). What type of function is it? Identify which is the independent variable and which is the dependent variable.

Calculate the exact coordinates of the axes intercepts of the function $h(x) = \frac{6}{x-2} + 3$.

Starting from the graph of the function $f(x) = \frac{1}{x}$, explain which geometric transformations (horizontal and vertical translations) must be applied to obtain the graph of the function $g(x) = \frac{1}{x-5} + 4$.

A portable swimming pool is filled in $12\text{ hours}$ using one hose. If we use $x$ identical hoses, the filling time is $y$ hours. Write the formula for the function modelling this situation and calculate how long it would take to fill the pool if we use $3$ hoses simultaneously.

Marcos is studying the function $f(x) = \dfrac{10}{x-4} + 3$, which models the average waiting time in minutes at a citizen service office depending on the number $x$ of open service desks. In his notebook, he writes the following conclusion:

"Since $x-4$ appears in the denominator, the indeterminacy arises when the denominator is zero with the sign changed, so the vertical asymptote is the line $x = -4$. Furthermore, since $3$ is added at the end, the horizontal asymptote is $y = 3$. Therefore, the domain of the function is $\mathbb{R} - \{-4\}$ and its center of symmetry is $(-4, 3)$".

1. a) Identify and explain with reasoning the conceptual error Marcos made regarding the vertical asymptote.
2. b) Write the corresponding correction, stating the true vertical asymptote, the correct domain of the function, and the real coordinates of its center of symmetry.

A freight delivery truck regularly travels the $360\text{ km}$ route along the motorway between Madrid and Valencia at a constant average speed $v$ (measured in $\text{km/h}$).

1. a) Write the algebraic expression of the function $t(v)$ that gives the travel time (in hours) as a function of the average speed $v$. State the constant of proportionality and justify whether the relationship between both quantities is direct or inverse.
2. b) If roadworks force the truck to reduce its average speed by exactly half, what happens to the travel time? Justify your answer using the algebraic expression of the function.
3. c) The Directorate-General for Traffic (DGT) establishes that trucks must travel on motorways at a minimum speed of $60\text{ km/h}$ and a maximum permitted speed of $90\text{ km/h}$. Determine the domain of definition of this function adapted to the real-world problem and find the interval of hours in which the travel time varies.

Express the rational function $f(x) = \frac{3x-7}{x-2}$ in the canonical form $f(x) = \frac{k}{x-a} + b$ by carrying out the corresponding polynomial division. Then, state the equations of its asymptotes and its domain.

Given the function $f(x) = \frac{-4}{x-1} + 2$:

1. a) State its domain and range.
2. b) Determine the equations of its asymptotes.
3. c) Graph the function on the coordinate axes.

You can do it on paper and upload a photo of your answer.

Boyle's Law states that, at constant temperature, the volume occupied by a gas is inversely proportional to the pressure to which it is subjected. A gas occupies a volume of $4\text{ litres}$ at a pressure of $1,5\text{ atmospheres}$.

Find the formula for the function relating volume $V$ to pressure $P$. What volume will the gas occupy if the pressure is increased to $3\text{ atmospheres}$?

A 4th ESO class is organising an end-of-year trip. Hiring the coach costs $540\text{ \euro}$.

Write the function that determines the price $p$ each student must pay as a function of the number $x$ of students attending. If $27$ students go in the end, how much will each pay? What if $3$ more students sign up at the last minute?

The graph shows how the price per student varies with the number of attendees:

Find algebraically the points of intersection between the hyperbola $y = \frac{4}{x}$ and the line $y = -x + 5$.

The following graph corresponds to a function of the form $f(x) = \frac{k}{x-a} + b$.

By observing the graph and its asymptotes, determine the value of the parameters $k$, $a$ and $b$.

Given the following piecewise-defined function:

$f(x) = \begin{cases} \frac{2}{x} & \text{if } x < -1 \\ 2x & \text{if } -1 \le x \le 2 \\ \frac{8}{x} & \text{if } x > 2 \end{cases}$

Study algebraically its continuity at the boundary points $x = -1$ and $x = 2$.

To print some promotional leaflets, a printer charges a fixed fee of $30\text{ \euro}$ for design and machine set-up, plus $0,05\text{ \euro}$ for each printed leaflet.

Write the function giving the average cost per leaflet as a function of the number $x$ of leaflets ordered. What value does the average cost tend to if a huge number of leaflets are printed?

The 4th year ESO students at a secondary school in Zaragoza rent a bus for a total fixed price of $720\text{ \euro}$ for a field trip to the Aragonese Pyrenees. The individual cost $y$ (in euros) that each student must pay is shared equally among the $x$ students who attend.

1. a) Estimate without performing exact numerical calculations: if $30$ students were initially planning to attend and $6$ people cancel at the last minute, will the price per person increase by around $1\text{ \euro}$, about $5\text{ \euro}$, or more than $15\text{ \euro}$? Briefly explain your intuitive estimate.
2. b) Model the situation by writing the function $y = f(x)$. Calculate analytically the price per person if $30$ students travel and if $24$ travel, and check whether your initial estimate was accurate.
3. c) A student states: "If we double the number of attendees, the price per person is halved; therefore, if a huge number of students came, the trip would end up being completely free". Using the concept of the horizontal asymptote of the function, explain why that statement is not mathematically valid in practice.

To determine the formula of a translated hyperbola that has the line $x = 3$ as its vertical asymptote, the line $y = 2$ as its horizontal asymptote, and whose graph passes through the point $(5, 6)$, two students in the group propose two different working methods:

• Laura's method: Directly use the standard form $f(x) = \dfrac{k}{x-h} + v$, identifying the values of $h$ and $v$ from the asymptotes and substituting the coordinates of the point $(5, 6)$ to solve for $k$.

• Daniel's method: Start from the general form $f(x) = \dfrac{ax+b}{x+c}$, substitute the point $(5, 6)$, look for two other approximate points on a freehand drawn graph, and solve the resulting system of three equations with three unknowns to calculate $a$, $b$, and $c$.

1. a) Compare both procedures, assessing their simplicity, accuracy, and speed. Justify which of the two methods is the most appropriate and efficient for solving the exercise.
2. b) Apply the chosen method to calculate the final algebraic expression of the function and write the coordinates of its center of symmetry.
3. c) Analytically check whether the point $(4, 10)$ belongs to the graph of that function by substituting its coordinates into the formula found.

A student has to analyse and sketch the rational function $f(x) = \frac{x^2 - x - 6}{x - 3}$. In their report, they conclude that the function has a vertical asymptote at $x=3$ because this value makes the denominator zero, and they present the following sketch:

Prove analytically why the student's statement is false, state what type of discontinuity the function actually has at $x=3$ and calculate the true limit at that point to correct the graph.

Determine analytically the points of intersection between the rational function $f(x) = \frac{2x+3}{x-1}$ and the line $g(x) = x+1$.

Find the values of the parameters $a$, $b$ and $c$ such that the function $f(x) = \frac{ax+b}{cx-2}$ has a horizontal asymptote at $y = 3$, a vertical asymptote at $x = -1$ and its graph passes through the origin.

A water tank has two filling taps. The first tap takes $x$ hours to fill it on its own, and the second tap takes $x+3$ hours to fill it on its own.

Express the time $t$ it would take to fill the tank if both taps are opened at the same time, in terms of $x$. If together they take exactly $2\text{ hours}$, how long does each tap take on its own?

A rectangular wooden frame is to be built with an interior area of $24\text{ cm}^2$.

Write the function relating the perimeter $P$ of the frame in terms of the length of its base $x$. Using the algebraic expression, determine whether the function has an oblique asymptote and explain its geometric meaning.

Discuss, in terms of the parameter $m$, the number of points of intersection between the hyperbola $y = \frac{-2}{x}$ and the line $y = mx + 2$.

Given the inverse proportion function $f(x) = \frac{x+2}{x-1}$:

Find the expression for the composite function $(f \circ f)(x)$, fully simplifying the result. From what you obtain, what conclusion can you draw about the inverse function of $f(x)$?

The total production cost of $x$ units of an electronic component in a factory is given by the function $C(x) = 2000 + 15x$.

Write the average cost function per unit $C_m(x) = \frac{C(x)}{x}$. Analyse whether this function has horizontal asymptotes and explain in detail its economic meaning in the context of the factory's production.

Solve the inequality $\frac{3x-1}{x+2} \le 2$ by studying the function $f(x) = \frac{3x-1}{x+2}$. To do this, obtain its canonical form $f(x) = \frac{k}{x-a} + b$ using polynomial division and determine the solution interval by observing the intersections.

Design the algebraic expression of a translated inverse proportionality function of the form $g(x) = \dfrac{k}{x-h} + v$ that simultaneously satisfies the following three conditions:

1. Its domain is $\mathbb{R} - \{2\}$.

2. Its range is $\mathbb{R} - \{-1\}$.

3. Its graph passes through the origin of coordinates, $(0, 0)$.

1. a) Deduce step by step the numerical values of the parameters $h$, $v$, and $k$, and write the final formula of the function.
2. b) Observe the sign of the parameter $k$ obtained. Explain, based on the relative position of the point $(0, 0)$ with respect to the center of symmetry $(h, v)$, why it was inevitable that the branches of the hyperbola would lie in the second and fourth quadrants relative to its asymptotes.
3. c) Analytically determine whether the curve intersects the coordinate axes at any point other than the origin $(0, 0)$ and justify your conclusion.

In a homeowners' association of $60$ dwellings in Sevilla, the installation of a shared photovoltaic system is approved to reduce electricity bills. The total installation cost is $18\,000\text{ \euro}$, which is shared equally among the dwellings that decide to participate. In addition, the maintenance company charges an annual fixed fee of $50\text{ \euro}$ to each participating dwelling for managing the service.

1. a) Model the function $C(x)$ that expresses the total cost for the first year (installation plus maintenance) borne by each dwelling as a function of the number $x$ of participating dwellings. Write the function in the form $C(x) = \dfrac{k}{x-h} + v$ and state the value of each parameter.
2. b) Determine the vertical and horizontal asymptotes of $C(x)$ and interpret their practical meaning in the context of the community in detail: what would happen to the individual cost if very few neighbors participated ($x \to 0$)? What value would the cost tend to if the community were infinitely large ($x \to \infty$)?
3. c) The board of owners sets a condition that the cost for the first year must not exceed $500\text{ \euro}$ per dwelling. Set up and solve an inequality based on your model to find the minimum number of neighbors needed for the project to go ahead.
4. d) Without calculating the final numerical amounts, reason: is the savings per neighbor when going from $10$ to $15$ participants greater than, equal to, or less than the savings obtained when going from $40$ to $45$ participants? Justify your answer based on the geometric shape and curvature of the hyperbola.