Inverse Variation Functions, Limits, and Continuity
Jayden and his friends want to rent a cabin for a weekend trip. It costs $\$240$, split equally. With 2 people, each pays $\$120$. With 10 people, each pays $\$24$. With 100 people, each pays just $\$2.40$. The more people, the less each one pays. But could the cost per person ever reach $\$0$? That relationship is inverse variation, and its graph is a two-branch curve that never touches the axes: a hyperbola. We'll answer the question at the end.
The Basic Function $f(x)=\dfrac{k}{x}$
In Algebra 1 you met inverse variation: two quantities whose product stays constant, $xy=k$. Solve for $y$ and you get a function. Some problems call it an inverse proportion function.
Why does it have two branches? Look at a table for $k=12$. When $x$ doubles, $y$ is cut in half, and negative inputs give negative outputs.
| $x$ | $-12$ | $-4$ | $-2$ | $-1$ | $1$ | $2$ | $4$ | $12$ |
|---|---|---|---|---|---|---|---|---|
| $f(x)=\dfrac{12}{x}$ | $-1$ | $-3$ | $-6$ | $-12$ | $12$ | $6$ | $3$ | $1$ |
If $k>0$, the branches lie in Quadrants I and III. If $k<0$, they lie in Quadrants II and IV. Drag the slider and watch.
Knowing that the function $f(x) = \frac{k}{x}$ passes through the point $(2, 6)$, find the value of the constant $k$ and calculate $\lim_{x \to 0^+} f(x)$.
Knowing that the function $f(x) = \frac{k}{x}$ passes through the point $(2, 6)$, find the value of the constant $k$ and calculate $\lim_{x \to 0^+} f(x)$.
Limits at Infinity and the Horizontal Asymptote
What happens to $\dfrac{12}{x}$ when $x$ gets enormous?
| $x$ | $10$ | $100$ | $1,000$ | $1,000,000$ |
|---|---|---|---|---|
| $\dfrac{12}{x}$ | $1.2$ | $0.12$ | $0.012$ | $0.000012$ |
Splitting a fixed amount $12$ into more and more parts makes each part smaller and smaller. The outputs get as close to $0$ as you like.
Calculate the limits of the function $g(x) = \frac{-5}{x}$ as $x \to +\infty$ and as $x \to -\infty$. Interpret the result geometrically.
Calculate the limits of the function $g(x) = \frac{-5}{x}$ as $x \to +\infty$ and as $x \to -\infty$. Interpret the result geometrically.
The time $t$ (in hours) it takes to travel a distance of $120$ miles as a function of the average speed $v$ (in mph) is given by $t(v) = \frac{120}{v}$. Calculate the…
The time $t$ (in hours) it takes to travel a distance of $120$ miles as a function of the average speed $v$ (in mph) is given by $t(v) = \frac{120}{v}$. Calculate the limit of $t(v)$ as $v \to +\infty$ and explain its physical meaning.
Boyle's Law states that, at a constant temperature, the volume $V$ occupied by a gas is inversely proportional to the pressure $P$ to which it is subjected, that is,…
Boyle's Law states that, at a constant temperature, the volume $V$ occupied by a gas is inversely proportional to the pressure $P$ to which it is subjected, that is, $V(P) = \frac{k}{P}$. If at a pressure of $1.5$ atmospheres the gas occupies a volume of $4$ gallons:
- a) Find the value of the constant $k$.
- b) Calculate $\lim_{P \to +\infty} V(P)$ and reason what happens to the gas if the pressure increases indefinitely.
One-Sided Limits and Vertical Asymptotes
Now go the other way: let $x$ get close to $0$. Try $\dfrac{12}{x}$:
| $x\to 0^+$ | $0.1$ | $0.01$ | $0.001$ |
|---|---|---|---|
| $\dfrac{12}{x}$ | $120$ | $1,200$ | $12,000$ |
| $x\to 0^-$ | $-0.1$ | $-0.01$ | $-0.001$ |
|---|---|---|---|
| $\dfrac{12}{x}$ | $-120$ | $-1,200$ | $-12,000$ |
Dividing by a tiny number gives a huge number, and the sign depends on the side you approach from.
To find the sign, plug in a number just beside $a$. For $x\to a^+$ the denominator $x-a$ is a tiny positive number, and for $x\to a^-$ it is a tiny negative number. Then use the rules for the sign of a quotient.
Given the inverse proportion function $f(x) = \frac{4}{x-3}$, determine its domain and calculate the one-sided limits as $x$ tends to $3$.
Given the inverse proportion function $f(x) = \frac{4}{x-3}$, determine its domain and calculate the one-sided limits as $x$ tends to $3$.
To organize an excursion, the cost of renting the bus is $600$ dollars, which will be shared equally among the attendees. The function that defines the cost per person…
To organize an excursion, the cost of renting the bus is $600$ dollars, which will be shared equally among the attendees. The function that defines the cost per person as a function of the number of attendees $x$ is $C(x) = \frac{600}{x}$.
Calculate $\lim_{x \to 0^+} C(x)$ and $\lim_{x \to +\infty} C(x)$. Interpret both results in the context of the problem.
The intensity of an acoustic signal $I$ (in watts per square foot) received by a microphone decreases in inverse proportion to the distance $d$ (in feet) from the…
The intensity of an acoustic signal $I$ (in watts per square foot) received by a microphone decreases in inverse proportion to the distance $d$ (in feet) from the speaker, following the model $I(d) = \frac{150}{d}$.
- a) Calculate $\lim_{d \to 0^+} I(d)$ and explain why the microphone cannot be placed right next to the speaker.
- b) Calculate $\lim_{d \to +\infty} I(d)$ and interpret the result.
Translated Hyperbolas
From the transformations you know that replacing $x$ with $x-h$ slides a graph $h$ units right, and adding $v$ slides it $v$ units up. Do that to a hyperbola and its asymptotes move too.
Determine the value of the following limits for the function $f(x) = \frac{1-x}{x}$: a) $\lim_{x \to 0^-} f(x)$ b) $\lim_{x \to 0^+} f(x)$ c) $\lim_{x \to +\infty} f(x)$
Determine the value of the following limits for the function $f(x) = \frac{1-x}{x}$:
- a) $\lim_{x \to 0^-} f(x)$
- b) $\lim_{x \to 0^+} f(x)$
- c) $\lim_{x \to +\infty} f(x)$
The temperature $T$ (in degrees Fahrenheit) of an electronic component as a function of the time $t$ (in minutes) since the cooling fan is turned on is given by…
The temperature $T$ (in degrees Fahrenheit) of an electronic component as a function of the time $t$ (in minutes) since the cooling fan is turned on is given by $T(t) = 25 + \frac{80}{t+2}$.
- a) What is the temperature at the instant the fan is turned on ($t=0$)?
- b) Calculate the limit of $T(t)$ as $t \to +\infty$. What does this value represent in reality?
From the basic inverse proportion function $f(x) = \frac{1}{x}$, a translation of $3$ units to the right and $2$ units upward is applied to obtain a new function $g(x)$.…
From the basic inverse proportion function $f(x) = \frac{1}{x}$, a translation of $3$ units to the right and $2$ units upward is applied to obtain a new function $g(x)$.
- a) Write the analytical expression of $g(x)$ in the form $\frac{ax+b}{cx+d}$.
- b) Prove, by calculating limits, that the asymptotes of $g(x)$ coincide with the displacements applied to the coordinate axes.
Rational Functions $\dfrac{ax+b}{cx+d}$
A translated hyperbola, written as one fraction, always looks like a linear expression over a linear expression. These are rational functions with a linear numerator and a linear denominator, and you can find their asymptotes directly from the coefficients.
You can also use these formulas "backwards" to find unknown parameters.
Find the equation of the horizontal asymptote and the vertical asymptote of the homographic function $h(x) = \frac{2x+1}{x-4}$.
Find the equation of the horizontal asymptote and the vertical asymptote of the homographic function $h(x) = \frac{2x+1}{x-4}$.
Given the function $f(x) = \frac{3x-1}{x+2}$: a) Calculate the points of intersection with the coordinate axes. b) Find the equations of its asymptotes by evaluating the…
Given the function $f(x) = \frac{3x-1}{x+2}$:
- a) Calculate the points of intersection with the coordinate axes.
- b) Find the equations of its asymptotes by evaluating the corresponding limits.
It is known that the function $f(x) = \frac{ax+2}{3x-b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$. Calculate the values of the…
It is known that the function $f(x) = \frac{ax+2}{3x-b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Calculate the values of the parameters $a$ and $b$.
Continuity and Types of Discontinuity
When one of the three conditions fails, you get one of three kinds of discontinuity:
-
Removable: the limit exists but $f(a)$ is missing or different. This is a hole.
-
Jump: the one-sided limits exist but are different.
-
Infinite: one-sided limits are $\pm\infty$ (a vertical asymptote).
Study the continuity of the function $f(x) = \frac{1}{x+5} - 2$. What type of discontinuity does it have at its singular point?
Study the continuity of the function $f(x) = \frac{1}{x+5} - 2$. What type of discontinuity does it have at its singular point?
Removable Discontinuities: When the Fraction Simplifies
Not every fraction with $x$ in the denominator is an inverse variation. If the numerator shares a factor with the denominator, the fraction simplifies and you get a hole instead of an asymptote. Use the value you get when you plug in $a$ to decide:
| Plugging in $x=a$ gives | Meaning |
|---|---|
| a number | continuous at $a$ |
| nonzero $\div\,0$ | vertical asymptote |
| $\dfrac00$ | indeterminate: factor and cancel, then take the limit |
Given the function $f(x) = \frac{3x-6}{x-2}$, calculate the limit as $x \to 2$. Take into account whether it is an indeterminate form or a vertical asymptote.
Given the function $f(x) = \frac{3x-6}{x-2}$, calculate the limit as $x \to 2$. Take into account whether it is an indeterminate form or a vertical asymptote.
Determine the domain and study the continuity of the function $f(x) = \frac{x^2-4}{x-2}$. Is it a pure inverse proportion function? Justify your answer by calculating…
Determine the domain and study the continuity of the function $f(x) = \frac{x^2-4}{x-2}$. Is it a pure inverse proportion function? Justify your answer by calculating the limit as $x \to 2$.
Piecewise Functions and Parameters
Sometimes a problem hands you a piecewise function with an unknown constant and asks you to make it continuous. The recipe comes straight from the definition: at the breaking point $x=a$, the left limit, the right limit, and $f(a)$ must all be equal.
Find the value of the parameter $k$ so that the piecewise function is continuous at $x = 1$:…
Find the value of the parameter $k$ so that the piecewise function is continuous at $x = 1$:
$f(x) = \begin{cases} \frac{k}{x+1} & \text{if } x < 1 \\ 2x - 3 & \text{if } x \ge 1 \end{cases}$
The piecewise function is defined as: $f(x) = \begin{cases} \frac{x+k}{x-2} & \text{if } x < 1 \\ x^2 - a & \text{if } x \ge 1 \end{cases}$ Determine the values of the…
The piecewise function is defined as:
$f(x) = \begin{cases} \frac{x+k}{x-2} & \text{if } x < 1 \\ x^2 - a & \text{if } x \ge 1 \end{cases}$
Determine the values of the constants $a$ and $k$ knowing that $\lim_{x \to -\infty} f(x) = 1$ and that the function is continuous at $x = 1$.
Inverse of a Rational Function and Its Asymptotes
To find an inverse, swap $x$ and $y$ and solve for $y$.
Given the function $f(x) = \frac{3x-2}{x+1}$: a) Find its inverse function $f^{-1}(x)$. b) Calculate the horizontal and vertical asymptotes of both $f(x)$ and…
Given the function $f(x) = \frac{3x-2}{x+1}$:
- a) Find its inverse function $f^{-1}(x)$.
- b) Calculate the horizontal and vertical asymptotes of both $f(x)$ and $f^{-1}(x)$.
- c) Explain the geometric and analytical relationship that exists between the asymptotes of a homographic function and those of its inverse.
Let the function be $f(x) = \frac{x+1}{x-1}$. a) Determine the analytical expression of the composite function $(f \circ f)(x) = f(f(x))$, simplifying the result as much…
Let the function be $f(x) = \frac{x+1}{x-1}$.
- a) Determine the analytical expression of the composite function $(f \circ f)(x) = f(f(x))$, simplifying the result as much as possible.
- b) In view of the result, what is the limit of $(f \circ f)(x)$ as $x \to 1$? Is there any contradiction with the domain of the original function? Reason your answer.
Modeling with Inverse Variation
Models of the form $\dfrac{c}{x}$ (possibly plus a constant) have two limits worth interpreting. The limit as $x\to0^+$ says what happens with a tiny input, and the limit as $x\to+\infty$ says what happens in the long run. When a model has several terms, take the limit of each term and combine the results.
Geometry gives another model. Suppose a rectangle must have a fixed area of $36$ square units. If the base is $x$, the height must be $\dfrac{36}{x}$, so the area stays $x\cdot\dfrac{36}{x}=36$. Its perimeter is a sum of two terms:
$$P(x)=2x+2\cdot\frac{36}{x}=2x+\frac{72}{x}$$| Base $x$ | $0.5$ | $1$ | $3$ | $6$ | $12$ | $36$ |
|---|---|---|---|---|---|---|
| Height $\frac{36}{x}$ | $72$ | $36$ | $12$ | $6$ | $3$ | $1$ |
| Perimeter $P(x)$ | $145$ | $74$ | $30$ | $24$ | $30$ | $74$ |
An electronic component manufacturing company has fixed monthly costs of $2000$ dollars and a variable cost of $15$ dollars for each unit produced. The average cost…
An electronic component manufacturing company has fixed monthly costs of $2000$ dollars and a variable cost of $15$ dollars for each unit produced. The average cost function per unit, if $x$ units are produced, is $C_m(x) = \frac{2000 + 15x}{x}$.
- a) Study the continuity of $C_m(x)$ in the context of the problem ($x > 0$).
- b) Calculate $\lim_{x \to +\infty} C_m(x)$ and prove analytically why the average cost per unit can never be exactly $15$ dollars, no matter how many units are manufactured.
We want to construct a rectangle with a fixed area equal to $36 \text{ cm}^2$. If we call $x$ the length of the base, the perimeter of the rectangle as a function of $x$…
We want to construct a rectangle with a fixed area equal to $36 \text{ cm}^2$. If we call $x$ the length of the base, the perimeter of the rectangle as a function of $x$ is given by $P(x) = 2x + \frac{72}{x}$.
- a) Calculate $\lim_{x \to 0^+} P(x)$ and $\lim_{x \to +\infty} P(x)$.
- b) Interpret geometrically what shape the rectangle takes in each of these two limiting cases and why the perimeter tends to infinity in both situations.
Saturation Models: Why Some Values Are Never Reached
Many real situations (learning, pollution cleanup, chemical reactions) follow a rational model with a horizontal asymptote as a ceiling or a vertical asymptote as a barrier. Use the tools you already have: the vertical asymptote comes from the denominator, the horizontal asymptote from the ratio of leading coefficients, and a one-sided limit is the right tool when the domain stops at the asymptote.
The cost $C$ (in thousands of dollars) to remove $p$ percent of the pollutants from a lake is modeled by the function $C(p) = \frac{120p}{100-p}$, valid for…
The cost $C$ (in thousands of dollars) to remove $p$ percent of the pollutants from a lake is modeled by the function $C(p) = \frac{120p}{100-p}$, valid for $0 \le p < 100$.
- a) Calculate the cost of removing $50\%$ and $90\%$ of the pollutants.
- b) Calculate $\lim_{p \to 100^-} C(p)$ and explain, from an economic and environmental point of view, why it is impossible to remove $100\%$ of the pollutants.
In biochemistry, Michaelis-Menten kinetics describes the rate of enzymatic reaction $v$ as a function of the substrate concentration $S$. The equation is…
In biochemistry, Michaelis-Menten kinetics describes the rate of enzymatic reaction $v$ as a function of the substrate concentration $S$. The equation is $v(S) = \frac{V_{max} \cdot S}{K_m + S}$, where $V_{max} = 100 \,\mu\text{mol/s}$ is the maximum rate and $K_m = 5 \text{ mM}$ is a constant.
- a) Calculate $\lim_{S \to +\infty} v(S)$ and state what $V_{max}$ represents graphically.
- b) Find the substrate concentration $S$ required for the rate of the reaction to be exactly half of its maximum asymptotic limit.
Back to the cabin: the cost per person is $C(x)=\dfrac{240}{x}$ dollars. Since $\displaystyle\lim_{x\to+\infty}\frac{240}{x}=0$, the price gets as close to $\$0$ as you like. But $y=0$ is a horizontal asymptote the curve never touches, because $240$ divided by any number is never $0$. The trip will never be totally free.