Information about a 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).

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 $f(x) = \frac{2x}{x-1}$, 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.

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.

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).

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:

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

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.

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.

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.

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).

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.