Exercises: Information about the function

Given the rational function $f(x) = \frac{x^2 + 3}{x^2 - 9}$, carry out a preliminary study of its basic elements by determining:

1. a) The domain of the function.

1. b) The points where it crosses the axes of coordinates.

1. c) The existence of symmetries (state whether it is even, odd or has no symmetry).

A team of engineers is monitoring the temperature of an electronic component. The evolution of the temperature $T$ (in degrees Celsius) with respect to time $t$ (in minutes) is given by the function:

$T(t) = \dfrac{40t + 100}{t + 2}$ with $t \geq 0$

You are asked to:

1. a) Find the initial temperature of the component ($t=0$).

1. b) Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem (long-term temperature).

Study the monotonicity (increasing and decreasing intervals) and calculate the relative extrema (maximums and minimums) of the following polynomial function:

$f(x) = x^3 - 3x + 2$

Given the function $f(x) = \frac{2x}{x-1}$, determine:

1. The equations of its vertical and horizontal asymptotes.

2. The position of the curve with respect to these asymptotes by studying the one-sided limits and limits at infinity.

Analyse the curvature of the function $f(x) = x^4 - 6x^2 + 5$. To do this:

1. Find the second derivative $f''(x)$.

2. Determine the intervals of concavity ($\cup$) and convexity ($\cap$).

3. Find the x-coordinates of the points of inflection.

The following information is known about a function $f(x)$ that is continuous on all of $\mathbb{R}$:

• It crosses the $X$ axis at $x = -2$ and $x = 2$.

• It has a relative minimum at $(0, -4)$.

• It is decreasing on $(-\infty, 0)$ and increasing on $(0, +\infty)$.

• $\lim_{x \to +\infty} f(x) = +\infty$ and $\lim_{x \to -\infty} f(x) = +\infty$.

Draw a sketch of the graph that satisfies all these conditions. To help you, here are the axes with the known points; copy the axes onto your paper and draw the curve.

You can do it on paper and send a photo with your answer.

Determine the period of the trigonometric function $f(x) = \sin(2x)$. Next, state with reasons whether the function exhibits any symmetry with respect to the origin or the y-axis.

Given the function $f(x) = \frac{x^2 - 1}{x}$, carry out a study of its infinite branches:

1. a) Check whether it has vertical asymptotes.

1. b) Check whether it has horizontal asymptotes.

1. c) If it has no horizontal asymptote, find out whether it has an oblique asymptote and find its equation.

In a 6th Year Leaving Certificate exam, Aoife has to analyze the rational function $f(x) = \frac{2x^2 - 8}{x^2 - 5x + 6}$ and writes the following in her working:

“1. The zeros of the denominator are $x = 2$ and $x = 3$. Therefore, the graph has two vertical asymptotes at the lines $x = 2$ and $x = 3$.

1. Since the function contains quadratic terms ($x^2$) and constant terms in both the numerator and the denominator, the function is even and symmetric with respect to the $OY$-axis”.

1. a) Identify the two conceptual errors made by Aoife and explain with mathematical rigor why her reasoning is incorrect in each case.
2. b) Carry out the appropriate correction: determine the real domain of $f$, factor the expression to simplify it, calculate the relevant one-sided limits to classify the behavior at $x = 2$ and $x = 3$ (distinguishing between a vertical asymptote and a removable discontinuity), and formally check the symmetry by evaluating $f(-x)$.

Consider the function $f(x) = \frac{4x - 1}{\sqrt{x^2 + 3}}$.

1. a) Estimate with reasoning: Before applying analytical limit techniques, reason about the behavior of the function when the variable $x$ takes extremely large values in absolute value ($x \to +\infty$ and $x \to -\infty$). From that qualitative intuition:

• Estimate how many horizontal asymptotes you expect to find and at what approximate values of $y$ they will be located.

• Predict whether the graph intersects the horizontal axis of abscissas ($OX$) and at what point it should do so.

1. b) Check formally: Calculate the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$, paying attention to the sign introduced by the square root when factoring out terms ($√{x^2} = |x|$), and deduce the exact equations of the horizontal asymptotes. Find the intersection point with the $OX$-axis and assess whether your initial estimates were correct or if you missed any sign change.

Carry out a complete study of the monotonicity and relative extrema of the rational function $f(x) = \frac{x^2 - 4x + 4}{x - 1}$. Determine precisely the intervals where the function is increasing and decreasing, as well as the coordinates of its maximums and minimums, if any.

Consider the polynomial function $f(x) = \frac{1}{4}x^4 - 2x^2 + 3$. Find the intervals of concavity and convexity, as well as the exact coordinates of its inflection points. Use the symbol $\cup$ to denote the intervals where the function is convex (curving upward) and $\cap$ for the intervals where it is concave (curving downward).

Determine all the asymptotes (vertical, horizontal and oblique) of the function $f(x) = \frac{2x^2 + 1}{x - 3}$. If an oblique asymptote exists, calculate its equation in the form $y = mx + n$ and study the position of the curve with respect to this asymptote as $x \to \infty$.

Sketch the graph of a continuous function $f(x)$ that simultaneously satisfies the following conditions obtained from its analytic study:

• Domain: $\mathbb{R} - \{0\}$

• Axes intercepts: $(2, 0)$ and $(-2, 0)$.

• Vertical asymptote at $x = 0$, with $\lim_{x \to 0^-} f(x) = -\infty$ and $\lim_{x \to 0^+} f(x) = +\infty$.

• Horizontal asymptotes: $y = -1$ as $x \to +\infty$ and $y = 1$ as $x \to -\infty$.

• $f'(x) < 0$ on its entire domain.

You can do it on paper and submit a photo with your answer.

A team of engineers monitors the charge level of a new lithium battery using the function $B(t) = \frac{100t}{t^2 + 9} + 10$, where $B$ is the percentage of charge and $t$ is the time in hours ($t \ge 0$).

1. a) At what time does the battery reach its maximum charge level? Justify that it is a maximum.

1. b) What value does the battery charge approach if it is left plugged in indefinitely? Justify your answer by calculating limits.

Given the graph of the derivative function $f'(x)$ shown below (a straight line passing through the points $(-2, 0)$ and $(0, 4)$), state with reasons:

1. a) The intervals where the original function $f(x)$ is increasing and decreasing.
2. b) The x-coordinate of the relative extrema of $f(x)$ and their type (maximum or minimum).
3. c) The interval of concavity or convexity of $f(x)$.

Consider the function $f(x) = \ln(x^2 + k)$, where $k$ is a real parameter.

1. a) Determine the value of $k$ so that the function has a point of inflection at $x = 1$.
2. b) For that value of $k$, study the domain of the function and check whether it exhibits any symmetry (even or odd).

Given the piecewise function:

$f(x) = \begin{cases} x^2 + 2x & \text{if } x \le 0 \\ \frac{ax}{x+1} & \text{if } x > 0 \end{cases}$

1. a) Calculate the value of $a$ so that the function is differentiable at $x = 0$.
2. b) For the obtained value of $a$, investigate whether the function has a horizontal asymptote as $x \to +\infty$.

Design the analytical expression of an irreducible rational function $f(x) = \frac{P(x)}{Q(x)}$ (where $P(x)$ and $Q(x)$ are polynomials with real coefficients) that simultaneously satisfies the following five conditions:

1. Its graph is symmetric with respect to the $OY$-axis (axis of ordinates).

2. Its domain of definition is $\mathbb{R} \setminus \{-2, 2\}$, having vertical asymptotes at both abscissas.

3. It has a horizontal asymptote with equation $y = 3$.

4. It intersects the vertical axis of ordinates at the point $(0, -1)$.

5. It does not intersect the axis of abscissas ($OX$) at any point.

Justify step by step how you choose the degree, the roots, and the coefficients of $P(x)$ and $Q(x)$ to satisfy each of the given conditions. Finally, formally verify that the resulting function has a relative extremum at $(0, -1)$ by studying the sign of its first derivative $f'(x)$.

Based on hydrological data inspired by reports from the Ministry for the Ecological Transition (MITECO), the percentage of dammed water capacity $V(t)$ in a river basin in the southern Iberian Peninsula over time $t$ (in months, with $t \ge 0$, where $t = 0$ represents the start of a special management plan following a drought) is modeled by the function:

1. a) Interpret the physical meaning of the initial value $V(0)$ in the context of the problem and justify whether it makes sense to look for intersection points with the axis of abscissas for $t \ge 0$.
2. b) Study the existence of horizontal asymptotes as $t \to +\infty$. Interpret their meaning in terms of long-term environmental management and explain why, according to this model, the volume of stored water will never reach $60\,\%$.
3. c) Analyze the monotonicity of $V(t)$ by calculating the sign of its first derivative for $t > 0$. Then, obtain the second derivative $V''(t)$ and determine in which exact month the filling rate of the reservoirs reaches its maximum speed (inflection point of the curve).

Consider the function $f(x) = \frac{ax^2 + bx}{x + c}$. Determine the values of the real parameters $a$, $b$, and $c$ knowing that the following conditions are simultaneously met:

1. The line $x = 2$ is a vertical asymptote of the graph of the function.

2. The function has a relative extremum at the point with abscissa $x = 3$.

3. The tangent line to the graph at the point of intersection with the $OY$ axis is parallel to the line $y = -2x + 5$.

Once the parameters are found, study the existence of an oblique asymptote and determine the relative position of the curve with respect to it.

The following graph corresponds to the derivative function $f'(x)$ of a continuous function $f(x)$ defined on all of $\mathbb{R}$. Using the visual information provided:

1. State the intervals where the original function $f(x)$ is increasing and decreasing.

2. Determine the x-coordinates of the relative extrema of $f(x)$, classifying them as maximums or minimums.

3. Study the curvature of $f(x)$ and locate its points of inflection.

4. Given that $f(0) = 1$, sketch the graph of the original function $f(x)$, justifying the shape obtained.

Given the function $f(x) = (x^2 - 1) \times e^{-x}$, carry out a comprehensive study for its graphical representation by analysing:

1. Domain and axis intercepts.

2. Behaviour at infinity (limits and asymptotes).

3. Monotonicity intervals and relative extrema (x- and y-coordinates).

4. Points of inflection and concavity/convexity.

5. Use all the above information to sketch the graph of the function, highlighting in particular the horizontal asymptotic behaviour.

A team of engineers is designing the profile of a jump ramp for a skateboarding competition. The side profile follows the curve of the function $f(x) = \ln(x^2 - 2x + 2)$ in the interval $[0, 3]$.

1. Find the point on the ramp that is at the lowest height above the ground.

2. Determine the points on the ramp where the slope is maximum and minimum. What physical interpretation do these points have in the context of skateboarding?

3. Investigate whether the function has any point of inflection in this interval and explain how the skater's sensation changes when passing through it.

Let the piecewise function be defined by:

$f(x) = \begin{cases} \frac{x^2 - kx}{x-1} & \text{if } x < 0 \\ \ln(x^2 + 1) + ax + b & \text{if } x \ge 0 \end{cases}$

1. Calculate the values of $a$ and $b$ so that the function is differentiable at $x = 0$, knowing that in the first piece the parameter $k$ takes the value $1$.

2. For the values found, investigate whether the function has an oblique asymptote as $x \to -\infty$. If so, find it.

3. Does the function have any relative extremum in the interval $(0, +\infty)$? Justify your answer.

We wish to inscribe a rectangle of maximum area in the region of the first quadrant bounded by the $OX$-axis, the $OY$-axis and the graph of the function $f(x) = 4e^{-0.5x}$.

1. Express the area of the rectangle as a function of the x-coordinate $x$ of one of its vertices located on the $OX$-axis.

2. Calculate the dimensions of this rectangle so that its area is maximum.

3. Draw a sketch of the situation using the following visual diagram.

You can do it on paper and send a photo with your answer.

Study the continuity and differentiability of the function $f(x) = x \times |x - 2|$. From this study:

1. Express the function as a piecewise defined function.

2. Calculate the intervals where the function is increasing and decreasing.

3. Determine the relative and absolute maximums and minimums in the interval $[0, 4]$.

4. Study the curvature and points of inflection. Pay special attention to the point where the function changes its definition.

Consider the function $f(x) = \frac{x^2 - 4}{x^2 - 1}$.

1. Analyse its symmetries and domain.

2. Determine all its asymptotes and the position of the graph with respect to them.

3. Find the points on the graph where the tangent line is horizontal.

4. Prove that the function is concave in its entire domain except in the interval $(-1, 1)$, where it is convex.

5. Sketch the graph of the function, combining all the previous elements.

You can do it on paper and send a photo with your answer.

To study the asymptotes and the relative position of the curve with respect to them for the rational function $f(x) = \frac{2x^3 - 3x^2 - 5x + 6}{x^2 - 1}$, two 6th Year (Leaving Certificate) students propose different methods:

• Method 1 (Standard limits): Directly calculate the zeros of the denominator ($x = \pm 1$) to propose vertical asymptotes, determine the oblique asymptote $y = mx + n$ using the limits $m = \lim_{x \to \infty} \frac{f(x)}{x}$ and $n = \lim_{x \to \infty} [f(x) - mx]$, and study the position of the curve with respect to the asymptote by evaluating the difference $f(x) - (mx + n)$ at large numerical values such as $x = 1000$ and $x = -1000$.

• Method 2 (Prior simplification and polynomial decomposition): Factor the numerator and denominator to simplify common factors before the asymptotic calculation, and perform Euclidean polynomial division to express the function in the form $f(x) = mx + n + \frac{R(x)}{D(x)}$.

1. a) Critically compare both methods: what false assumption is made by someone applying Method 1 at $x = 1$? Why does evaluating at specific values ($x = \pm 1000$) not constitute a rigorous proof of the position of the curve?
2. b) Justify with reasoning which of the two methods is safer, more general, and more efficient for this type of function.
3. c) Following the most convenient method, determine with justification the only vertical asymptote, the equation of the oblique asymptote, the exact relative position of the curve at $\pm\infty$ using the algebraic sign of the remainder, and the coordinates of the point where there is a removable discontinuity.

Consider a continuous function $f$ on $\mathbb{R} \setminus \{-1\}$ whose derivative is:

1. a) Study in detail the sign of $f'(x)$ and deduce the intervals of increase and decrease of $f$, as well as the $x$-coordinates of all its relative extrema. Justify with reasoning why at $x = 1$ the function does not have a relative extremum despite $f'(1) = 0$ holding.
2. b) Knowing that the curvature of $f$ depends on the growth of the derivative function itself, calculate the second derivative $f''(x)$ simplifying the expression as much as possible. Determine the intervals of concavity ($f'' < 0$) and convexity ($f'' > 0$) of $f$, find the $x$-coordinates of its inflection points, and classify precisely the nature of the singular point located at $x = 1$.
3. c) Furthermore, knowing that $\lim_{x \to -1^+} f(x) = -\infty$ and that $f(1) = 4$, prove rigorously that the graph of $f$ cannot intersect the line $y = 4$ at any point in the interval $(1, +\infty)$.