Functions of a Real Variable and Types of Functions

Classify the following functions according to their type (polynomial, rational, radical, exponential or logarithmic):

1. a) $f(x) = 3^x - 1$

1. b) $g(x) = \frac{x^2 + 1}{x - 2}$

1. c) $h(x) = \log_{10}(x^2)$

1. d) $i(x) = x^3 - 5x + 2$

1. e) $j(x) = \sqrt[3]{x - 4}$

Determine the domain of definition of the following functions of a real variable based on the restrictions of the operations that form them:

1. a) $f(x) = \frac{2x - 5}{x^2 - 9}$

1. b) $g(x) = \sqrt{3x + 12}$

1. c) $h(x) = \ln(x - 1)$

Find the widest possible domain of definition for the following real function of a real variable, expressing the result as a union of intervals:

To solve it, you must simultaneously consider the restrictions imposed by the square root, the natural logarithm and the denominator of the fraction.

Given the absolute value function $f(x) = |x - 3|$, represent it as a piecewise function and calculate its value for $x = 0$ and $x = 5$.

A telephone company offers a data plan where the first $10$ GB cost a fixed $\$15$. From then on, each extra GB is charged at $\$2$. The function representing the total cost $C(x)$ in terms of the GB consumed $x$ is:

$C(x) = \begin{cases} 15 & \text{if } 0 \le x \le 10 \\ 15 + 2(x - 10) & \text{if } x > 10 \end{cases}$

Study whether the function is continuous at the critical point $x = 10$.

A new video game streaming service applies a monthly rate based on high-speed data consumption. The first $20$ GB are included in a fixed fee of $\$15$. From then on, the price for each additional GB increases linearly according to the function $P(x) = 15 + 0,5(x - 20)$, where $x$ is the total number of GB consumed.

However, if the user exceeds $100$ GB, the company applies a penalty for excess traffic and the function becomes $P(x) = 0,1x^2 - 45$.

Write the analytical expression of the total cost function $C(x)$ as a piecewise function and check if the price "jumps" or is continuous exactly at the moment of consuming $100$ GB.

Given the functions $f(x) = x + 3$ and $g(x) = x^2 - 1$, perform the following operations and determine the domain of the resulting function:

1. a) $(f + g)(x)$

1. b) $(f \cdot g)(x)$

1. c) $(\frac{f}{g})(x)$

Consider the functions $f(x) = \sqrt{x + 2}$ and $g(x) = \frac{1}{x^2 - 1}$.

1. Find the expression of the composite function $(g \circ f)(x)$.

2. Determine the domain of $(g \circ f)(x)$.

3. Does this domain coincide with that of the function $h(x) = \frac{1}{x+1}$? Justify your answer.

An aerospace engineering company is designing the ascent trajectory of a probe. The height $h$ in miles, measured from a reference level (negative below it), with respect to time $t$ in minutes is given by a composite function. It is known that the outer function is $f(x) = \frac{x-2}{x+3}$ and the inner function is $g(t) = \sqrt{t^2 - 1}$.

Calculate the analytical expression of the total height function $H(t) = (f \circ g)(t)$ and determine its domain and range with reasoning in the physical context of the problem, knowing that the probe begins its mission at $t = 1$.

Find the inverse function $f^{-1}(x)$ of the function $f(x) = \frac{e^x + 3}{2}$. State the domain and range of both functions and graphically verify their symmetry with respect to the line $y = x$.

An artificial intelligence algorithm processes input data $x$ using a modified logistic activation function:

We want to find the inverse function $A^{-1}(x)$ so that the system can perform the reverse process (decoding). Find this inverse and determine for which values of the output $y$ decoding is possible (range of the original function).

Let the function be $f(x) = \frac{3^x - 1}{3^x + 1}$. Prove analytically that the function is one-to-one in its entire domain. Then, find the expression of its inverse function $f^{-1}(x)$ and clearly specify the domain and range of this inverse.

Study the symmetry (even, odd or asymmetrical) of the following functions, justifying the steps by calculating $f(-x)$:

1. $f(x) = \frac{x^3}{x^2 + 1}$

2. $g(x) = x \cdot \cos(x)$

3. $h(x) = \frac{e^x + e^{-x}}{2}$ (Hyperbolic cosine function)

Consider the family of functions $f_k(x) = \frac{x^2 - k^2}{x^2 + k^2}$ where $k \in \mathbb{R}, k \neq 0$.

1. Prove that for any value of $k$, the function is even.

2. Find the points of intersection with the axes in terms of $k$.

3. Find the value of $k$ so that the function takes the value $y = 1/2$ when $x = 3$.

4. Discuss whether there is any value of $k$ for which the function has vertical asymptotes.

Every function $f(x)$ can be uniquely decomposed, at the points $x$ where both $f(x)$ and $f(-x)$ exist, as the sum of an even function $g(x)$ and an odd function $h(x)$, such that:

Given the function $f(x) = \frac{x+2}{x-1}$, find its even and odd components. Prove analytically that, indeed, $g(x)$ is an even function and $h(x)$ is an odd function. What are the domains of each of these three functions?

In the study of sound signals, the function $s(t) = \ln(t)$ is used to represent the intensity of a primary signal. Due to interference, the resulting signal is transformed according to the rule $g(t) = |s(|t-2|)-1|$.

Graphically represent the final function $g(t)$ starting from the elementary function $\ln(t)$ and explaining in detail each of the geometric transformations (translations, reflections and moduli) that take place.

From the following graph of the function $f(x)$, identify the values of the one-sided limits and the image of the function at the point of discontinuity $x = 1$:

1. a) $\lim_{x \to 1^-} f(x)$

1. b) $\lim_{x \to 1^+} f(x)$

1. c) $f(1)$

Observe the following graph and visually identify the equations of its vertical and horizontal asymptotes:

Calculate the following direct limits by substitution, checking if the value obtained is finite:

1. a) $\lim_{x \to 2} (x^2 - 3x + 1)$

1. b) $\lim_{x \to 0} \frac{e^x + 1}{x + 3}$

1. c) $\lim_{x \to -1} \sqrt{x + 5}$

Given the rational function $f(x) = \frac{2x^2 - 8}{x^2 - x - 2}$:

1. Calculate its domain.

2. Identify if there is any point where the function has a "removable discontinuity" (a point where the numerator and denominator are simultaneously zero).

3. Calculate the points of intersection with the coordinate axes.

A surveillance drone carries out a test flight following a trajectory whose height in feet with respect to time in seconds is given by the function:

1. From what height does the drone take off?

2. What is the maximum height it reaches and in which second does this occur?

3. If the drone must land when its battery reaches $5\%$ (which happens at $21$ seconds), at what height will it be at that moment? Explain whether the result makes physical sense.

Given the piecewise function:

Find the values of the real parameters $a$ and $b$ knowing that the function passes through the point $(-2, 0)$ and has a removable discontinuity at the point with x-coordinate $x = 1$. Once found, study the symmetry of the function in the interval $[1, +\infty)$.