Exercises: Rational Functions

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

Find the coordinates of the x- and y-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 indicating the intervals of the 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 algebraically whether the function $f(x) = \frac{3x}{x^2+1}$ is even, odd, or has no symmetry with respect to the coordinate axes.

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

The average production cost (in dollars) 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 toward which the average cost tends if production becomes infinitely large, interpreting the result in the context of the problem.

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

A tank initially contains 500 gallons of pure water. A saline solution with a concentration of 40 g/L begins to be pumped in at a rate of 10 gal/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 $T(x)$ they take if they work 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 the younger worker have taken to paint the wall by himself?

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

Find the equation of the inverse function of $f(x) = \frac{2x-1}{x+3}$ and specify what the domain of this new function $f^{-1}(x)$ is.

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

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

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

Discuss, in terms of 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 completely lacks vertical asymptotes.

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

The concentration $C(t)$ of a drug in the bloodstream $t$ hours after its intramuscular administration is given by the expression $C(t) = \frac{30t}{t^2+9}$ (measured in mg/L). Without using calculus, prove purely algebraically that the maximum concentration can never exceed $5$ mg/L and determine at what exact time $t$ 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 analyze the condition that its discriminant must satisfy for the time to be a real number).

Let the rational function be $f(x) = \frac{x+2}{x-1}$. Prove algebraically 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 one point. Interpret geometrically what these specific lines represent for the curve.

The population of a fish species 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 as time passes. (Hint: Try to express the fraction in the form $A - \frac{B}{t^2+1}$ by performing polynomial division).