Exercises: Types of functions

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 points of intersection with the coordinate axes.
2. b) Find the increasing and decreasing intervals using the first derivative $f'(x)$.
3. c) Determine the $(x, y)$-coordinates of its relative maxima and minima.

Consider the rational function $f(x) = \frac{2x + 4}{x - 3}$. To understand its structure before representing it, analyze its main elements:

1. a) Determine its domain of definition.
2. b) Calculate the equations of its vertical and horizontal asymptotes.
3. c) Study the symmetry of the function. Is it even, odd, or does it have no symmetry?

Absolute value functions often have points where the graph changes direction abruptly. Let the function be $f(x) = |x - 4|$.

1. a) Express the function as a piecewise function.
2. b) Indicate the point on the graph where the function is continuous but not differentiable (corner point).
3. c) Sketch its graph indicating the vertex.

The domain of a function is fundamental for its representation. 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 there are restrictions in each case.

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 ($t \geq 0$).

1. a) What is the population at the start of the experiment ($t = 0$)?
2. b) What is the limit of the population as time increases indefinitely? Interpret the result in terms of asymptotes.
3. c) Does the population increase or decrease over time? Justify your answer using the derivative $P'(t)$.

Given the function $f(x) = x^4 - 2x^2$, study its curvature by carrying out the following 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 parity of the following functions, justifying mathematically whether they are even ($f(x) = f(-x)$), odd ($f(-x) = -f(x)$), or neither:

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

Observe the following graph of a function $f(x)$ and extract the requested information:

1. a) State the values of $x$ where the vertical asymptotes are located.
2. b) Estimate the intervals where the function is increasing.
3. c) Identify the value of the $\lim_{x \to \infty} f(x)$ by observing the behavior of the graph.

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. Find the domain of definition and the points of intersection with the coordinate axes.

2. Study the existence of symmetries in the function.

3. Determine the vertical and horizontal asymptotes and the position of the curve with respect to them.

4. Calculate the increasing and decreasing intervals, and determine its relative maxima and minima.

5. Sketch the graph of the function based on the previous results.

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

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

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

3. Use the information on the increase and extrema to draw an accurate sketch of the graph.

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 since the start of the experiment ($t \geq 0$).

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

2. What value does the population tend to if the experiment is prolonged indefinitely? Justify your answer by calculating limits.

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

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

1. Express the function as a piecewise function.

2. Study the differentiability of the function at the points $x = 1$ and $x = 3$. What happens geometrically at these points?

3. Find the relative extrema of the function and represent it graphically.

Let the function be $f(x) = (x - 2)e^x$. Carry out a detailed study of it to represent it graphically, including:

1. Domain and points of intersection with the axes.

2. Horizontal asymptotes and behavior at infinity.

3. Intervals of monotonicity and relative extrema.

4. Points of inflection and curvature.

Study and represent 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 representation?

2. Determination of the increasing and decreasing intervals.

3. Study of the curvature and search for points of inflection.

4. Does the function have asymptotes of any kind? Justify 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 piece in its corresponding interval.

4. Sketch the graph 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 a point of inflection at $x = 1$ and the slope of the tangent line at that point is $1$. Once the parameters have been found:

1. Study the monotonicity of the resulting function.

2. Determine if it has any relative maximum or minimum.

3. Justify whether the function has asymptotes.

Consider the piecewise function:

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

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 the differentiability at $x = -2$ and $x = 0$.

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

4. Determine the increasing and decreasing intervals, as well as the relative maxima and minima in the interval $(-\infty, 0)$.

Let the rational function be 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. Once the parameters have been calculated:

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

2. Find the points of intersection with the axes.

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

A communications satellite follows 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 the tracking ($t \ge 0$).

1. Determine the domain of the function and identify the instants when the signal is completely interrupted.

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

3. Calculate the increasing and decreasing intervals of the signal intensity.

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

Given the function:

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

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

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

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

4. Determine the increasing and decreasing intervals and the relative extrema.

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

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

1. Calculate the points of intersection with the axes and the asymptotes of the function.

2. Determine the intervals of monotonicity and the local extrema (maxima and minima).

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.

Study and graphically represent the "mixed type" function:

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

To do this, analyze in detail:

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

2. The vertical and oblique asymptotes.

3. The existence of relative extrema using the first derivative.

4. The concavity and convexity through the study of the sign of $f''(x)$.

A ramp profile for a skatepark is to be designed. The side profile follows the curve of the function:

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

for $x \in [-5, 5], x \neq 0$. Answer the following:

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

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 can 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 gradient is zero.

5. Study whether there are horizontal asymptotes when the domain is extended to the entire real line.

Draw a sketch of 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 $y = x$ when $x \to +\infty$.

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

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

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

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

• It intersects the $OX$-axis only at the points $x = -4$ and $x = 2$.