Rational Function

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

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

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 $x = 2$.

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

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.

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

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

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.

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

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 $x \to -\infty$.

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.

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.

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

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

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 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 $y = \frac{5}{3}$.

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 $(4, 15)$.

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 average production cost (in pesos) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{50\,000 + 300x}{x}$. Calculate the average cost of manufacturing 100 units and determine the value toward which the average cost tends as production becomes infinitely large, interpreting the result in the context of the problem.

A tank initially contains 500 liters 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?

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?

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).

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).