Intercepts, Sign, and Asymptotes of Rational Functions

Calculate the points of intersection with the coordinate axes of the function $f(x) = \frac{2x-6}{x+4}$.

Determine the intervals in which the function $f(x) = \frac{x-3}{x+2}$ takes strictly positive values.

Study analytically the sign of the function $f(x) = \frac{-2x}{x^2-4}$ throughout its domain.

Find the domain of definition and the equations of the vertical asymptotes of the function $f(x) = \frac{x+7}{x^2-9}$.

Discuss, depending on the values of the real parameter $k$, the number and type of vertical asymptotes that the function $f(x) = \frac{x^2-x-2}{x^2-k}$ has. Also determine for what exact value of $k$ the function has a removable discontinuity.

Given the function $f(x) = \frac{3x^2-2}{x^2+1}$, determine the equation of its horizontal asymptote by evaluating the relevant limits as $x \to +\infty$ and $x \to -\infty$.

Find the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{5x+1}{2x-4}$.

The population of an algae species in a pond, measured in thousands of individuals, evolves according to the function $P(t) = \frac{120t+50}{2t+5}$, where $t \geq 0$ is the time elapsed in days. Calculate the initial population, the instant when the population reaches $40$ thousand individuals and the limit of the population in the very long term (horizontal asymptote).

Calculate the equation of the oblique asymptote of the function $f(x) = \frac{x^2-5x+6}{x-1}$.

The total cost of producing $x$ hundreds of pieces of an electronic component is $C(x) = 3x^2+5x+12$ hundreds of dollars. If the average cost function per hundred pieces is defined as $C_m(x) = \frac{C(x)}{x}$, find the oblique asymptote of $C_m(x)$ and explain its economic meaning for very large production values.

Calculate the equation of the oblique asymptote of the function $f(x) = \frac{2x^2+1}{x-1}$ and study analytically the relative position of the curve with respect to this asymptote as $x \to +\infty$ and as $x \to -\infty$.

Observe the following graph of a rational function and determine, from it, the equations of its asymptotes and the coordinates of its points of intersection with the axes.

From the graph of the rational function $f(x)$ shown below, deduce its analytical expression knowing that it is of the form $f(x) = \frac{ax+b}{cx+d}$.

The following graph shows a rational function that has an oblique asymptote and a vertical asymptote. Knowing that the function is of the form $f(x) = \frac{x^2+ax+b}{x+c}$, analyze its visual elements (asymptotes and intersections) to determine the value of the coefficients $a$, $b$ and $c$.

Determine the numerical values of the parameters $a$ and $b$ knowing that the function $f(x) = \frac{ax+3}{bx-2}$ has a vertical asymptote at the line $x = 1$ and a horizontal asymptote at the line $y = 4$.

Find the values of the parameters $a$, $b$ and $c$ so that the rational function $f(x) = \frac{ax^2+bx+c}{x+2}$ has an oblique asymptote with equation $y = 3x - 1$ and crosses the x-axis exactly at the point $x = 1$.

Given the rational function $f(x) = \frac{x^3+kx^2+x-1}{x^2-1}$, find the value of the parameter $k$ so that the representative curve of the function crosses its own oblique asymptote at the point with x-coordinate $x = 2$.

Find the asymptotes and the points of intersection with the axes of the function $f(x) = \frac{x^3-8}{x^2-4}$. Pay special attention to possible removable discontinuities when simplifying the expression.

The deflection of a supported beam, subjected to a variable load, can be approximated by the function $D(x) = \frac{x^3-4x^2+3x}{x^2+1}$, where $x$ is the distance in feet measured from the left end of the support. Determine the points where the deflection is zero, study the sign of $D(x)$ in the interval $x \in [0, 4]$ and calculate the equation of the oblique asymptote of the function.

Study completely the asymptotes and the relative position of the curve with respect to them for the function $f(x) = \frac{2x^3-5x^2+4x+1}{x^2-2x+1}$. Take into account the possible multiple roots that may appear in the denominator during the calculation of limits.

The concentration of a polluting gas in a test chamber is given by the expression $C(t) = \frac{5t}{t^2+4}$ parts per million (ppm), with $t \geq 0$ measured in hours. Determine at what instants the concentration is zero, study the sign of $C(t)$ to interpret its physical meaning, and find the horizontal asymptote indicating what happens in the long term.

The amount of a medication present in the bloodstream, in milligrams, after $t$ hours of its oral administration is modeled by the function $M(t) = \frac{50t^2}{t^3+8}$. Study the points of intersection with the axes, the sign of the function for $t \geq 0$ and its asymptotic behavior.

The efficiency of a modified thermal cycle, as a function of a compression ratio $v > 1$, is given by the model $E(v) = \frac{v^2-3v+2}{v^2+v}$. Analyze the sign of the efficiency to determine in what range of $v$ the cycle produces useful work (that is, when $E(v) > 0$) and find the theoretical limiting efficiency as $v \to +\infty$.