Exercises: Functions of a real variable. Types of functions

Sergio has to determine the domain of the composite function $(f \circ g)(x)$, where $f(x) = x^2$ and $g(x) = \sqrt{2x - 6}$. On his exam, he writes the following:

"First, I calculate the analytical expression:

Since the result is a first-degree polynomial and polynomials exist for any real number, the domain of $(f \circ g)$ is $\mathbb{R}$."

1. a) Point out the conceptual error Sergio makes when applying the definition of a composite function.
2. b) Explain why the domain of the composition cannot be $\mathbb{R}$ and find the correct domain of $(f \circ g)$.
3. c) If you were to plot the complete line $y = 2x - 6$ and the actual graph of $(f \circ g)(x)$ on Cartesian axes, how would they differ visually?

A public car park in downtown Málaga applies a short-stay rate that depends on the parking duration $t$ (in minutes) according to the following conditions during the first 3 hours ($0 \le t \le 180$):

• During the first 20 minutes ($0 \le t \le 20$), a fixed entry fee of $0,80$ € is charged, regardless of the exact number of minutes used in that interval.

• From minute 20 up to 120 minutes ($20 < t \le 120$), it is billed at a rate of $0,03$ € for each minute elapsed after minute 20, which is added to the initial $0,80$ €.

• After two hours ($120 < t \le 180$), a fixed flat rate of $4,50$ € is charged.

1. a) Write the expression for the price function $P(t)$ (in euros) as a piecewise-defined function for $t \in [0, 180]$.
2. b) Verify analytically whether the function is continuous at the transition points between intervals ($t = 20$ and $t = 120$) by calculating the corresponding one-sided limits.
3. c) Interpret the result in economic terms: does the user experience an abrupt jump in the charge at either of these two moments, or does the cost evolve continuously?

Design the algebraic formula of a single function $f(x)$ (not defined piecewise) that simultaneously satisfies the following three conditions:

1. Its exact domain of definition is the set $\operatorname{Dom}(f) = [-6, -2) \cup (2, 6]$.

2. It is an even function; that is, its graph is symmetric with respect to the $y$-axis ($f(-x) = f(x)$).

3. Its formula is constructed using operations (quotient or product) combining a square root and another algebraic expression.

Describe step by step how you derive the expression for the numerator and the denominator to generate exactly the square brackets at $\pm 6$ and the parentheses at $\pm 2$. Finally, prove analytically that the function you constructed is even by verifying that $f(-x) = f(x)$.

Consider the function $f(x) = \ln\left(\dfrac{8 - x}{x + 3}\right)$.

1. Estimate and explore:

a) Before formally solving any inequality, mentally evaluate the sign of the quotient for the test values $x = -5$, $x = 0$, and $x = 10$. For which of them is $f(x)$ well-defined, and for which is it not?

b) Based on that exploration, estimate whether the domain will be an open and bounded interval $(a, b)$ or an unbounded half-line extending to infinity. What values do you guess its endpoints will be?

1. Calculate: Determine the domain of $f(x)$ with analytical rigor by setting up the inequality required by the logarithm and studying the signs of the numerator and the denominator.

2. Check and reflect: Compare the obtained domain with your estimate. Next, reason what happens to the value of $f(x)$ as $x$ approaches $8$ from the left ($x \to 8^-$) and as it approaches $-3$ from the right ($x \to -3^+$).

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.

An aerospace engineering company is designing the ascent trajectory of a probe. The height $h$ in kilometres, 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 $t = 1$.

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.

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 $[1, +\infty)$.

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.

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.

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.

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?

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 $15$ €. Beyond that, 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 an excess traffic penalty and the function becomes $P(x) = 0,1x^2 - 45$.

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 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.

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.

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

The graph of the trajectory is as follows:

1. a) From what height does the drone take off?

1. b) What is the maximum height it reaches and at which second does it occur?

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

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.

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

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

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

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) $f(1)$

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

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

A phone 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 $x = 10$.

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) $(\frac{f}{g})(x)$

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

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

Given the rational functions $f(x) = \dfrac{1}{x - 4}$ and $g(x) = \dfrac{2x + 1}{x - 3}$, you are asked to find the domain of the composite function $(f \circ g)(x)$. Two students argue for different approaches:

• Marcos's method: First, he simplifies the analytical expression:

Looking at the final fraction, he concludes that the only problematic value is the one that makes its denominator zero: $-2x + 13 = 0 \implies x = 6,5$. Therefore, he claims that $\operatorname{Dom}(f \circ g) = \mathbb{R} \setminus \{6,5\}$.

• Laura's method: She applies the theoretical definition of the domain of a composition:

She first requires that $x \in \operatorname{Dom}(g)$ ($x \ne 3$) and, subsequently, that $g(x) \ne 4$, solving $\dfrac{2x + 1}{x - 3} \ne 4 \implies x \ne 6,5$. She concludes that $\operatorname{Dom}(f \circ g) = \mathbb{R} \setminus \{3; 6,5\}$.

1. Determine which of the two methods is mathematically correct. Explain precisely which algebraic step taken by Marcos caused the loss of a domain restriction.

2. If a calculator were to evaluate $(f \circ g)(3)$ following the natural order of composite operations (first calculating $g(3)$ and then applying $f$ to the result), what would happen? Does this agree with what Marcos's final formula predicts?

3. Formulate a methodological conclusion: why can simplifying the algebraic expression of a function before finding its domain lead to serious errors?

Consider a function $f: \mathbb{R} \to \mathbb{R}$ defined piecewise by:

where $k$ is a real constant and $g(x)$ is a function to be determined in each case.

1. Even case: Find the algebraic expression that $g(x)$ must have so that $f$ is an even function on all of $\mathbb{R}$. Once found, analyze whether the continuity of $f$ at $x = 0$ depends on the value of the parameter $k$, justifying it by computing the one-sided limits.

2. Odd case and proof:

a) Prove algebraically that any function $h: \mathbb{R} \to \mathbb{R}$ that is odd and defined at the origin necessarily satisfies $h(0) = 0$.

b) Find the algebraic expression for $g(x)$ so that $f$ satisfies the odd condition $f(-x) = -f(x)$ for all $x \ne 0$.

c) Using the proof from part (a), deduce what value the constant $k$ must necessarily take so that the function $f$ is odd on all of $\mathbb{R}$ and, at the same time, continuous at the transition point $x = 0$.

Unit challenge. Analyze the thermal dynamics of your municipality to optimize home heating and cooling energy costs.

1. Data and real numbers: Access the AEMET website and download the temperatures of the last $24$ hours at your nearest station. Express the daily range using a bounded interval $[T_{\min}, T_{\max}] \subset \mathbb{R}$, round to tenths, and determine the bound on the absolute error made.

2. Probability: Using the series from the last $10$ days, calculate the frequentist probability that the temperature exceeds $21,5\text{ }^\circ\text{C}$. Justify why Laplace's rule is not applicable in this context and reflect on the assignment of probabilities to meteorological phenomena.

3. Modeling and derivative: In GeoGebra or in a spreadsheet, plot the scatter plot and perform a polynomial fit $T(t)$. Analytically find the derivative function $T'(t)$, determine the hours of maximum warming and cooling rates (extrema of the derivative), and compare the instantaneous rate of change $T'(10)$ with the average rate of change on the interval $[8, 12]$.

4. Deliverable: Design a one-page executive report with the graphs and a justified hourly ventilation recommendation.

The following will be assessed:

• Rigorous use of real numbers, intervals, and error estimation.

• Critical understanding of the frequentist versus classical interpretation.

• Correct calculation of the derivative function and interpretation of its physical meaning.