Exercises: Sign of a function. Axis intercepts and asymptotes of a rational function

Calculate the axis intercepts 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.

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

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

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

Observe the following graph of a rational function and, from it, determine the equations of its asymptotes and the coordinates of its axis intercepts.

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 $x \to -\infty$.

Analytically study the sign of the function $f(x) = \frac{-2x}{x^2-4}$ over its entire domain.

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 moment at which the population reaches $40$ thousand individuals, and the long-term limit of the population (horizontal asymptote).

The concentration of a pollutant 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 the moments at which the concentration is zero, study the sign of $C(t)$ to interpret its physical meaning, and find the horizontal asymptote stating what happens in the long term.

The total cost of producing $x$ hundred pieces of an electronic component is $C(x) = 60x^2+100x+240$ hundred pesos. 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 levels of production.

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 $y = 4$.

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

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

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

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

Find the values of the parameters $a$, $b$ and $c$ such 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 at exactly the point $x = 1$.

The deflection of a supported beam under 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 meters 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.

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

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

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

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

Fully investigate 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 any potential multiple roots that may appear in the denominator when calculating the limits.

A demographic model establishes that the growth rate of a population, measured as an annual percentage, is given by the function $T(x) = \frac{x^2-8x+15}{x^2+4}$, where $x \geq 0$ is the time elapsed in decades. Study the sign of $T(x)$ to determine the historical periods of population growth and decline, and calculate the horizontal asymptote to interpret the limiting long-term trend of the rate.