Functions of a Real Variable. Types of Functions

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

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

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

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

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 largest possible domain for the following real-valued function of a real variable, expressing the result as a union of intervals:

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

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

1. Find its domain.

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

3. Find the axis intercepts.

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

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 $g(t) = |s(|t-2|)-1|$.

Sketch the graph of the final function $g(t)$ starting from the basic function $\ln(t)$ and explaining in detail each of the geometric transformations (translations, reflections and modulus operations) that take place.

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

A surveillance drone carries out a test flight following a trajectory whose height in meters 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 it occur?

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

A phone company offers a data plan where the first $10$ GB cost a fixed $300$ pesos. From then on, each extra GB is charged at $40$ pesos. The function representing the total cost $C(x)$ in terms of the GB consumed $x$ is:

$C(x) = \begin{cases} 300 & \text{if } 0 \le x \le 10 \\ 300 + 40(x - 10) & \text{if } x > 10 \end{cases}$

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

A new video game streaming service applies a monthly fee based on high-speed data usage. The first $20$ GB are included in a fixed charge of $300$ pesos. Beyond that, the price for each additional GB increases linearly according to the function $P(x) = 300 + 10(x - 20)$, where $x$ is the total number of GB consumed.

However, if the user exceeds $100$ GB, the company applies an excess traffic penalty and the function becomes $P(x) = 2x^2 - 900$.

Write the algebraic expression for the total cost function $C(x)$ as a piecewise-defined function and check whether the price "jumps" or is continuous right at the point of consuming $100$ GB.

Given the piecewise function:

Find the values of the real parameters $a$ and $b$ knowing that the function passes through point $(-2, 0)$ and that it is continuous at the point of abscissa $x = 1$. Once found, study the symmetry of the function on the interval $[1, +\infty)$.

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) $(f \times g)(x)$

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 for 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 kilometers, 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 $g(t) = \sqrt{t^2 - 1}$.

Find the algebraic expression of the total height function $H(t) = (f \circ g)(t)$ and give reasons for its domain and range 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 verify graphically their symmetry with respect to the line y = x.

An artificial intelligence algorithm processes input data $x$ by means of a modified logistic activation function:

The figure shows the graph of $A(x)$ for $x$ between $-10$ and $10$.

We want to find the inverse function $A^{-1}(x)$ so that the system can carry out the reverse process (decoding).

1. a) Find the expression of the inverse $A^{-1}(x)$.

1. b) Determine for which output values $y$ it is possible to perform the decoding (range of the original function) and justify your answer.

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

Study the symmetry (even, odd or neither) of the following functions, justifying the steps by evaluating $f(-x)$:

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

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

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

2. Determine the axis intercepts in terms of $k$.

3. Find the value of $k$ such 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.

Any 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. Verify algebraically that $g(x)$ is indeed an even function and $h(x)$ is an odd function. What are the domains of each of these three functions?

Calculate the following direct limits by substitution, checking whether 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) $\lim_{x \to -1} \sqrt{x + 5}$

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 $x = 1$. In the graph, the empty circle indicates a point that does not belong to the graph and the filled circle indicates a point that does belong to it.

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

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

1. c) $f(1)$

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