Types of functions

The domain of a function is fundamental for sketching its graph. Find the domain of the following transcendental functions:

1. a) $f(x) = \ln(x^2 - 1)$
2. b) $g(x) = e^{\frac{1}{x}}$
3. c) $h(x) = \sqrt{x + 5}$

Briefly explain why restrictions exist in each case.

Determine the parity of the following functions, justifying mathematically whether they are even ($f(x) = f(-x)$), odd ($f(-x) = -f(x)$), or neither:

1. a) $f(x) = x^2 \times \cos(x)$
2. b) $g(x) = \frac{x^3}{x^2 + 1}$
3. c) $h(x) = e^x - x$

Look at the following graph of a function $f(x)$ (the dashed lines are the asymptotes of the function; use the tick marks on the axes, one per unit) and extract the requested information:

1. a) State the values of $x$ where the vertical asymptotes are located.

1. b) Estimate the intervals where the function is increasing.

1. c) Identify the value of $\lim_{x \to +\infty} f(x)$ by observing the behavior of the graph.

Sketch the graph of a continuous function $f(x)$ that simultaneously satisfies the following eight conditions:

• Its domain is $\mathbb{R} - \{1\}$.

• It has a vertical asymptote at $x = 1$ such that $\lim_{x \to 1^-} f(x) = +\infty$ and $\lim_{x \to 1^+} f(x) = -\infty$.

• It has an oblique asymptote $y = x$ as $x \to +\infty$.

• It has a horizontal asymptote $y = -2$ as $x \to -\infty$.

• $f'(x) < 0$ on the interval $(-\infty, -2)$ and $f'(x) > 0$ on $(-2, 1) \cup (1, +\infty)$.

• It has a relative minimum at the point $(-2, -4)$.

• $f''(x) > 0$ on $(-3, 1)$ and $f''(x) < 0$ on $(-\infty, -3) \cup (1, +\infty)$.

• It intersects the $x$-axis only at the points $x = 0$ and $x = 2$.

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

Given the polynomial function $f(x) = x^3 - 3x^2 + 4$, carry out a basic study of its behavior following these steps:

1. a) Calculate the axis intercepts.
2. b) Find the intervals of increase and decrease using the first derivative $f'(x)$.
3. c) Determine the coordinates $(x, y)$ of its relative maxima and minima.

Given the function $f(x) = x^4 - 2x^2$, study its curvature by following these steps:

1. a) Calculate the second derivative $f''(x)$.
2. b) Find the intervals of concavity ($\cup$) and convexity ($\cap$).
3. c) Find the points of inflection of the function.

Determine the values of the parameters $a$ and $b$ so that the function $f(x) = ax^3 + bx^2 - x + 4$ has an inflection point at $x = 1$ and the slope of the tangent line at that point is $1$. Once you have found the parameters:

1. a) Study the monotonicity of the resulting function.

1. b) Determine whether it has any relative maximum or minimum.

1. c) Justify whether the function has asymptotes.

Consider the rational function $f(x) = \frac{2x + 4}{x - 3}$. To understand its structure before graphing it, analyze its main elements. The figure shows part of its graph; the tick marks on the axes are 1 unit apart.

1. a) Determine its domain.

1. b) Calculate the equations of its vertical and horizontal asymptotes.

1. c) Study the symmetry of the function. Is it even, odd, or does it show no symmetry?

Given the rational function $f(x) = \frac{2x^2}{x^2 - 9}$, carry out a complete study for its graphical representation following these steps:

1. a) Find the domain of definition and the points where the graph crosses the coordinate axes.

1. b) Study whether the function has any symmetries.

1. c) Determine the vertical and horizontal asymptotes and the position of the curve with respect to them.

1. d) Calculate the intervals of increase and decrease, and determine the relative maxima and minima.

1. e) Sketch the graph of the function based on the previous results.

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

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

1. Find the equations of its asymptotes. If an oblique asymptote exists, determine the position of the curve with respect to it as $x \to +\infty$ and $x \to -\infty$.

2. Find the intervals of concavity and convexity, and check whether there are points of inflection.

3. Use the information on increasing/decreasing intervals and extrema to draw an accurate sketch of the graph.

A biologist studies the evolution of a bacteria population in a controlled environment. The number of bacteria (in thousands) is given by the function $P(t) = \frac{40t}{t + 2}$, where $t$ is the time in hours since the start of the experiment ($t \geq 0$). The figure shows the graph of $P(t)$.

1. a) What is the population at the start of the experiment ($t = 0$)?

1. b) What is the limit of the population as time grows indefinitely? Interpret the result in terms of asymptotes.

1. c) Does the population increase or decrease over time? Justify your answer using the derivative $P'(t)$.

A biological study on an insect population in a controlled environment follows the function $P(t) = \frac{50t}{t^2 + 4} + 2$, where $P$ is the population in thousands and $t$ is the time in months from the start of the experiment ($t \geq 0$).

1. Determine at what time the population reaches its maximum value and what that value is.

2. What value does the population approach if the experiment were to continue indefinitely? Justify your answer by evaluating limits.

3. Study the curvature of the function to identify when the population decline begins to slow down (point of inflection).

Consider the rational function defined by:

$$f(x) = \frac{x^3 + ax^2 + b}{x^2 - 1}$$

Determine the values of the real parameters $a$ and $b$ knowing that the line $y = x + 2$ is an oblique asymptote of the function and that its graph passes through the point $(0, 0)$. Once the parameters are calculated:

1. Find the domain and the vertical asymptotes, studying the position of the curve relative to them.

2. Find the intercepts with the coordinate axes.

3. Study the existence of inflection points. Is it possible for this function to have no inflection points in its domain? Justify your answer.

Functions with absolute values often have points where the graph changes direction abruptly. Let $f(x) = |x - 4|$ be a function.

1. a) Express the function as a piecewise-defined function.

1. b) State the point on the graph where the function is continuous but not differentiable (corner point).

1. c) Sketch its graph, indicating the vertex.

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

Given the absolute value function $f(x) = |x^2 - 4x + 3|$:

1. a) Express the function as a piecewise-defined function.

1. b) Study the differentiability of the function at the points $x = 1$ and $x = 3$. What happens geometrically at those points?

1. c) Find the relative extrema of the function and graph it.

Given the function:

$$f(x) = \frac{x \times |x|}{x - 1}$$

1. Express the function as a piecewise defined function by removing the modulus.

2. Study the continuity and differentiability of the function in its entire domain, paying special attention to the point $x = 0$.

3. Find the asymptotes of the graph of $f$.

4. Determine the intervals where the function is increasing and decreasing and the relative extrema.

5. Sketch the graph of the function, clearly indicating the points where the derivative does not exist, if any.

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

Let $f(x) = (x - 2)e^x$ be the function. Carry out a detailed study of it in order to sketch its graph, including:

1. Domain and axis intercepts.

2. Horizontal asymptotes and behavior at infinity.

3. Intervals of monotonicity and relative extrema.

4. Points of inflection and curvature.

Analyse and sketch the logarithmic function $f(x) = \ln(x^2 + 1)$. In your analysis, you must include:

1. Proof that the function is even. How does this affect its graphical representation?

2. Determination of the intervals of increase and decrease.

3. Study of the curvature and finding the points of inflection.

4. Does the function have any type of asymptotes? Justify your answer.

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

A communications satellite describes a trajectory whose signal intensity received on Earth is given by the function:

$$S(t) = \ln |t^2 - 9| - \frac{t}{2}$$

where $t$ represents the time in hours since the start of tracking ($t \ge 0$).

1. Determine the domain of the function and identify the times at which the signal is completely interrupted.

2. Find the vertical asymptotes and explain their meaning in this context.

3. Calculate the intervals where the signal intensity is increasing and decreasing.

4. Determine at what time the signal reaches its relative maximum value and sketch the graph of the intensity trajectory.

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

Consider the piecewise function:

$$f(x) = \begin{cases} \frac{1}{x-2} & \text{if } x < 0 \\ x^2 - 2x - \frac{1}{2} & \text{if } x \geq 0 \end{cases}$$

1. Study the continuity and differentiability of the function at $x = 0$.

2. Find the asymptotes of the function in its domain of definition.

3. Calculate the local minima and maxima of each section in its corresponding interval.

4. Sketch the graph of the function.

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

Consider the piecewise-defined function:

$$f(x) = \begin{cases} \dfrac{x^2 - 4}{x+2} & \text{if } x < -2 \\ a \times e^{x+2} + b x & \text{if } -2 \le x < 0 \\ \dfrac{x+1}{x^2+1} & \text{if } x \ge 0 \end{cases}$$

Carry out a complete study of the function following these steps:

1. Find the values of $a$ and $b$ so that the function is continuous at $x = -2$ and has a relative extremum at $x = -1$.

2. For the values found, study differentiability at $x = -2$ and $x = 0$.

3. Calculate the asymptotes of the function on its entire domain.

4. Determine the intervals of increase and decrease, as well as the relative maxima and minima on the interval $(-\infty, 0)$.

Consider the function $f(x) = (x-2)^2 \times e^x$.

1. Calculate the axis intercepts and the asymptotes of the function.

2. Determine the intervals of monotonicity and local extrema (maximums and minimums).

3. Find the intervals of curvature and the points of inflection.

4. Knowing that $e^{-2} \approx 0.135$ and $e^0 = 1$, sketch the graph of the function.

5. Calculate the area of the triangle formed by the origin of coordinates and the points on the graph where the tangent is horizontal.

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

Study and graph the "mixed-type" function:

$$f(x) = x \times e^{1/x}$$

To do so, analyze in detail:

1. a) The domain of definition and the behavior of the function near $x = 0$, calculating the one-sided limits $\lim_{x \to 0^+} f(x)$ and $\lim_{x \to 0^-} f(x)$.

1. b) The vertical and oblique asymptotes.

1. c) The existence of relative extrema using the first derivative.

1. d) The concavity and convexity by studying the sign of $f''(x)$.

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

We wish to design the profile of a ramp for a skatepark. The side profile follows the curve of the function:

$$f(x) = \frac{\ln(x^2+1)}{x}$$

for $x \in [-5, 5], x \neq 0$. Find:

1. Show that the function is odd. What does this imply for its graph?

2. Define the value of $f(0)$ so that the function is continuous at $x=0$. To do this, calculate $\lim_{x \to 0} f(x)$ (you may use L'Hôpital's rule).

3. Is the function differentiable at $x=0$ with the previous definition? Justify your answer.

4. Determine the points on the ramp where the slope is zero.

5. Investigate whether there are horizontal asymptotes when the domain is extended to the whole real line.