Exercises: Rational function

Calculate the domain of definition of the rational function $f(x) = \frac{5x-2}{x^2-9}$.

Find the coordinates of the axis intercepts of the function $f(x) = \frac{3x+12}{2x-5}$.

Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{4x-1}{x+3}$.

Study the sign of the rational function $f(x) = \frac{x-4}{x+2}$, clearly stating the intervals of the real number line where the function takes positive values and where it takes negative values.

Find the value of $x$ for which the function $f(x) = \frac{2x+3}{x-1}$ takes exactly the value $f(x) = 5$.

Given the function $f(x) = \frac{1}{x+2} - 4$, explain what elementary geometric transformations (horizontal and vertical translations) have been applied to the graph of the basic hyperbola $y = \frac{1}{x}$ to obtain it.

Check analytically whether the function $f(x) = \frac{3x}{x^2+1}$ is even, odd or exhibits no symmetry with respect to the coordinate axes.

Determine the domain of the function $f(x) = \frac{x^2+1}{x^2+4x+4}$.

Aoife states the following in class: "For the rational function $f(x) = \frac{x^2 - 9}{x^2 - x - 6}$, since the values that make the denominator zero are $x = 3$ and $x = -2$, the graph necessarily has two vertical asymptotes: the lines $x = 3$ and $x = -2$". a) Identify the conceptual error Aoife makes and explain in detail why making the denominator zero is not sufficient on its own to guarantee a vertical asymptote. b) Correct her analysis by determining what actually happens at $x = 3$ and at $x = -2$ (distinguishing between a vertical asymptote and a removable discontinuity with a hole). c) Write the equations of all the asymptotes (vertical and horizontal) that the function actually possesses.

A translated inverse proportionality function has the canonical form $f(x) = \frac{k}{x - h} + v$, with $k \neq 0$. a) Find the values of the parameters $h$, $v$, and $k$ so that the function has the lines $x = -2$ and $y = 3$ as asymptotes, and its graph intersects the y-axis at the point $(0, 1)$. b) From the expression obtained, state the coordinates of its center of symmetry and calculate the exact x-intercept ($y = 0$). c) Justify with reasoning the sign of the function in the three intervals defined by its vertical asymptote and its root using a sign chart.

The average production cost (in euros) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{2500 + 15x}{x}$. Calculate the average cost of manufacturing 100 units and determine the value towards which the average cost tends as production becomes infinitely large, interpreting the result in the context of the problem.

Find the equation of the oblique asymptote of the rational function $f(x) = \frac{2x^2-5x+4}{x-1}$.

A tank initially contains 500 litres of pure water. A saline solution with a concentration of 40 g/L is pumped in at a rate of 10 L/min. The salt concentration in the tank after $t$ minutes is given by the function $C(t) = \frac{400t}{500+10t}$. At what time will the concentration be exactly 25 g/L? What is the equation of the horizontal asymptote of this function and what physical meaning does it have?

Determine the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{ax+7}{3x+b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = \frac{5}{3}$.

Two workers work together to paint a wall. The younger worker would take $x$ hours to do it working alone, while the older worker would take $x+3$ hours. The total time $T(x)$ it takes them when working simultaneously is given by the function $T(x) = \frac{x(x+3)}{2x+3}$. If working together they manage to finish the job in exactly 2 hours, how long would it have taken the younger worker to paint the wall on his own?

Find the domain, the axis intercepts, and the equations of all asymptotes of the rational function $f(x) = \frac{3x^2}{x^2-16}$.

Calculate the algebraic expression for the inverse function of $f(x) = \frac{2x-1}{x+3}$ and specify the domain of this new function $f^{-1}(x)$.

Given the function $f(x) = \frac{x^2+1}{x}$, determine its domain, its asymptotes (vertical, horizontal, or oblique), and study the position of the curve with respect to its oblique asymptote as $x \to +\infty$ and as $x \to -\infty$.

Consider the rational function $f(x) = \frac{6x - 5}{2x + 4}$. a) Estimate without calculating: For extraordinarily large values of $x$ in absolute value (such as $x = 1\,000\,000$ or $x = -1\,000\,000$), what integer value will $f(x)$ approach? Justify your prediction by analyzing the relative weight of the terms in the numerator and the denominator. b) Estimate the one-sided sign: Without performing complex divisions, deduce only the sign of the numerator and the denominator for a value slightly greater than $-2$ (for example, $x = -1.99$) and for one slightly less (for example, $x = -2.01$). Towards $+\infty$ or towards $-\infty$, where will the function shoot off on each side? c) Verify formally: Rigorously calculate the horizontal asymptote and the vertical asymptote using the corresponding limits: $\lim_{x \to \pm\infty} f(x)$, $\lim_{x \to -2^+} f(x)$, and $\lim_{x \to -2^-} f(x)$. Assess whether your previous intuitive estimates were correct.

A freight train makes the round trip between Dublin and the port of Cork (one-way distance: $300\text{ km}$). On the outbound journey, it travels at a constant average speed $v$ (in $\text{km/h}$). On the return journey, due to freight restrictions of the railway infrastructure, it is forced to reduce its average speed by $20\text{ km/h}$ compared to the outbound trip. a) Formulate and simplify into a single algebraic fraction the function $T(v)$ that expresses the total time taken for the complete trip (round trip) as a function of $v$. b) Determine the mathematically admissible domain of $T(v)$ and compare it with the domain that makes real physical sense for this route. c) Model the overall average speed of the complete trip, defined as $V_m(v) = \frac{600}{T(v)}$, obtaining a simplified rational function. d) Analyze and interpret in the real context of the trip the asymptotic behavior: what happens to the total time if $v \to +\infty$, and what happens if the outbound speed approaches $20\text{ km/h}$?

Find the values of the parameters $a$, $b$ and $c$ of the rational function $f(x) = \frac{ax^2+bx+c}{x-3}$ given that it has an oblique asymptote with equation $y = 2x+1$ and that its graph passes through the point with coordinates $(4, 15)$.

Analyse in detail the rational function $f(x) = \frac{x^2-5x+6}{x^2-4}$. Determine its domain, simplify the algebraic expression stating for which values the simplification is valid, find the equations of its asymptotes, and explain what graphical feature occurs at exactly the point with x-coordinate $x = 2$.

Discuss, depending on the real parameter $k$, the existence of vertical asymptotes for the function $f(x) = \frac{x^2-2x+5}{x^2-kx+9}$. Determine the interval of values of $k$ for which the function has no vertical asymptotes at all.

From the following graph of a rational function, determine its domain, the equations of its asymptotes (vertical and horizontal), and the axis intercepts. Next, suggest an algebraic expression of the type $f(x) = \frac{ax^2}{x^2+b}$ that perfectly matches all these visual features.

The concentration $C(t)$ of a drug in the bloodstream $t$ hours after intramuscular administration is given by the expression $C(t) = \frac{30t}{t^2+9}$ (measured in mg/L). Without using differentiation tools, prove algebraically that the maximum concentration can never exceed $5$ mg/L and determine the exact time $t$ at which this maximum concentration is reached. (Hint: Set up the equality $C = \frac{30t}{t^2+9}$ as a quadratic equation in the variable $t$ and analyse the condition that its discriminant must satisfy for the time to be a real number).

Let $f(x) = \frac{x+2}{x-1}$ be a rational function. Prove analytically that the composite function $(f \circ f)(x) = x$ for every value of $x$ belonging to its domain. What geometric conclusion about the graph of $f(x)$ can you draw from this result in relation to the line $y = x$?

Determine the values of the real parameter $m$ for which the line with equation $y = mx$ intersects the rational curve $y = \frac{1}{x-2}$ at exactly a single point. Geometrically interpret what these specific lines represent for the curve.

The population of a species of fish in a protected lake follows a growth model given by the function $P(t) = \frac{2000t^2+500}{t^2+1}$, where $t \ge 0$ is measured in years from the start of the study. Determine the initial population, the population in the long term (limit as $t \to +\infty$) and prove algebraically that the population is always growing over time. (Hint: Try to express the fraction in the form $A - \frac{B}{t^2+1}$ by carrying out polynomial division).

The goal is to calculate the oblique asymptote $y = mx + n$ of the rational function $f(x) = \frac{2x^3 - x^2 + 4}{x^2 + 2}$ and to determine whether the curve lies above or below said asymptote at $+\infty$ and at $-\infty$. Two procedures are proposed: Method A (Analytical limits), calculating $m = \lim_{x \to \infty} \frac{f(x)}{x}$ and $n = \lim_{x \to \infty} [f(x) - mx]$, and then evaluating the difference $f(x) - (mx + n)$ at specific numerical values (such as $x = 100$ and $x = -100$); and Method B (Euclidean division), performing polynomial division to express $f(x) = (mx + n) + \frac{R(x)}{Q(x)}$ and deducing the asymptote and the relative position analytically from the remainder. a) Apply both methods to find the equation of the oblique asymptote line. b) Using Method B, prove with algebraic rigor (without numerical trial and error) the position of the curve relative to the asymptote as $x \to +\infty$ and as $x \to -\infty$. c) Critically compare both methods: explain why Method B is more direct and conclusive for rational functions, and reason in what situations or with what type of non-polynomial functions Method B becomes inapplicable and resorting to Method A becomes essential.

Design the analytical expression of a rational function $f(x) = \frac{P(x)}{Q(x)}$ that simultaneously satisfies the following conditions: 1. Its domain is $\text{Dom}(f) = \mathbb{R} \setminus \{-1, 2\}$. 2. At $x = 2$ it presents a vertical asymptote, but at $x = -1$ it has a removable discontinuity ($\lim_{x \to -1} f(x)$ exists and is finite; that is, a hole). 3. Its only horizontal asymptote is the line $y = 3$. 4. The graph of the function intersects its own horizontal asymptote at the point with abscissa $x = 4$. 5. It passes through the y-intercept $(0, 6)$. a) Prove with reasoning why it is not possible that, after simplifying the common factor responsible for the removable discontinuity, the remaining denominator is of degree 1, deducing the minimum essential degree that $P(x)$ and $Q(x)$ must have to allow the curve to intersect its horizontal asymptote without adding more vertical asymptotes to the domain. b) Obtain the polynomials $P(x)$ and $Q(x)$ that satisfy all the conditions and formally verify each of them.