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.

To study the rational function $f(x) = \frac{x^2 - 4}{x^2 - 3x + 2}$, Lucía states in her exam:

"The denominator vanishes at $x = 1$ and at $x = 2$, so the graph has two vertical asymptotes: the lines $x = 1$ and $x = 2$. On the other hand, since the numerator vanishes at $x = 2$ and at $x = -2$, the function intersects the $X$-axis at the points $(2, 0)$ and $(-2, 0)$".

1. Point out the two conceptual errors made by Lucía and explain why her reasoning is incorrect based on the definitions of domain and vertical asymptote.

2. Correct the exercise: find the domain of definition of $f(x)$, its true vertical asymptotes (justifying using one-sided limits), and all its intersection points with the coordinate axes.

Given the rational function $f(x) = \frac{4 - 2x}{x + 3}$:

1. Estimate the sign of the function at the ends of the real line: before performing detailed calculations, observe the dominant terms of the numerator and denominator and predict whether for very large values of $x$ (such as $x = +1000$) and very negative values (such as $x = -1000$) the function will take positive or negative values. From that observation, estimate the equation of its horizontal asymptote.

2. Analytically find the intersection points with the coordinate axes and the equations of the asymptotes (horizontal and vertical).

3. Construct the sign chart of $f(x)$ by decomposing the real line using its critical points, and check whether your initial estimate about the behavior at the ends agrees with the obtained positive and negative regions.

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 $C(x) = 3x^2+5x+12$ hundred euros. 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 behaviour.

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.

An express courier company using electric vans in the metropolitan area of Valencia models the average operational cost per kilometer traveled, measured in euros per kilometer (€/km), as a function of the daily kilometers $x$ driven by each vehicle using the rational function:

1. Justify why the mathematical model is not defined for $x = 5$ and interpret what happens to the average cost when a van travels a distance very close to $5$ km, calculating the one-sided limit $\lim_{x \to 5^+} C(x)$ and stating the vertical asymptote.

2. Find the horizontal asymptote of the function using $\lim_{x \to +\infty} C(x)$ and interpret its economic meaning for the company on days with high-mileage routes.

3. Reasonably prove whether the average cost per kilometer can ever become less than or equal to $0,18$ €/km by analyzing the sign of the expression $C(x) - 0,18$ on the interval $(5, +\infty)$, and explain whether the graph intersects its horizontal asymptote.

Design the simplified algebraic expression of an irreducible rational function of the form $f(x) = \frac{P(x)}{Q(x)}$ that simultaneously satisfies the following four conditions:

1. It has exactly two vertical asymptotes: the lines $x = -2$ and $x = 3$.

2. Its only horizontal asymptote is the line $y = 2$.

3. It intersects the $X$-axis at a single point: $(1, 0)$.

4. The curve is tangent to the $X$-axis at that intersection point; that is, it does not change sign when crossing $x = 1$ (the function keeps the same sign immediately to the left and to the right of $x = 1$).

Explain how you determine the degrees of $P(x)$ and $Q(x)$, their roots, and their multiplicities. Finally, calculate the coordinates of the intersection point with the $Y$-axis and draw up the complete sign chart of your function, checking that all conditions are met.

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 metres 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.

Consider the rational function $f(x) = \frac{2x^3 - 3x^2 - 8x + 5}{x^2 - 4}$. To calculate its oblique asymptote $y = mx + n$ and determine the relative position of the curve with respect to that line as $x \to +\infty$ and as $x \to -\infty$, two working methods are proposed:

• Method A (Limit formulas): Calculate $m = \lim_{x \to \pm\infty} \frac{f(x)}{x}$ and $n = \lim_{x \to \pm\infty} (f(x) - mx)$, and then evaluate the difference $f(100) - (m \cdot 100 + n)$ using a calculator to estimate whether the curve lies above or below.

• Method B (Euclidean polynomial division): Express $f(x) = (mx + n) + \frac{R(x)}{Q(x)}$ by dividing $P(x)$ by $Q(x)$, and deduce the relative position by formally analyzing the sign of the fraction $\frac{R(x)}{Q(x)}$ at the ends.

You are asked to:

1. Apply Method B: perform the polynomial division step by step, write the equation of the oblique asymptote, and prove analytically (without resorting to tables of values or a calculator) whether the curve approaches from above or from below the asymptote at $+\infty$ and at $-\infty$.

2. Justify why Method B is theoretically superior and more informative than Method A for rational functions.

3. Determine whether the graph ever intersects its oblique asymptote at any point with a finite abscissa, by setting up and solving the corresponding equation from the obtained decomposition.

Consider the family of rational functions $f_k(x) = \frac{x^2 - kx + 4}{x^2 - 4}$, where $k \in \mathbb{R}$ is a real parameter.

1. Estimate and conjecture: Knowing that the denominator vanishes at $x = 2$ and at $x = -2$, reason before calculating whether you think it is possible to find values of $k$ for which the function has exactly two vertical asymptotes, exactly one, or none. Formulate a hypothesis based on the relationship between the roots of the numerator and the denominator.

2. Analytical verification:

a) Determine the exact values of $k$ for which the function has a removable discontinuity instead of a vertical asymptote. For those values, state the equation of the single surviving vertical asymptote and calculate the coordinates $(x, y)$ of the "hole" in the graph.

b) For the particular case $k = 5$, find analytically the intersection points with the coordinate axes, all its asymptotes, and the sign of the function in each of the intervals into which $\mathbb{R}$ is divided.

c) For the particular case $k = 0$, analyze whether the graph intersects its horizontal asymptote and conclude whether your initial hypothesis about the possibility of the function having no vertical asymptotes at all was correct or impossible in $\mathbb{R}$.