Exercises: Domain; range; symmetry; periodicity; intercepts and sign

A smart device records the battery level of a mobile phone over a morning. The function $B(t)$ represents the charge percentage as a function of time $t$ (in hours) elapsed since 8:00 am.

Observing the graph, state:

1. The domain of the function (recorded time interval).

2. The range (minimum and maximum battery values).

3. At what times does the battery level remain constant?

Determine the domain of the following analytical functions:

1. $f(x) = \frac{x + 5}{x - 4}$

2. $g(x) = x^2 - 3x + 2$

3. $h(x) = \sqrt{x - 3}$

Find the axis intercepts for the quadratic function representing the profit of a small T-shirt shop:

$B(x) = -x^2 + 6x - 5$

(Where $x$ is the price in euros and $B(x)$ is the profit in hundreds of euros).

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

1. $f(x) = x^4 + 3x^2$

2. $g(x) = x^3 - x$

3. $h(x) = \frac{2}{x}$

Given the following graph representing the breathing rate of an athlete at rest (volume of air in litres against time in seconds), identify with clear reasoning whether the function is periodic and, if so, state its period.

Study the sign of the linear function $f(x) = 3x - 9$. To do so:

1. Calculate the point where the function equals zero (intercept with the $X$-axis).

2. State in which interval of $x$ the function is positive ($f(x) > 0$) and in which it is negative ($f(x) < 0$).

Determine the domain of the following logarithmic functions, remembering that the argument must be strictly greater than zero:

1. $f(x) = \log(x + 2)$

2. $g(x) = \ln(6 - 2x)$

From the graph of the function $f(x)$ shown below, determine:

1. a) The value of the intercepts with the $Y$ axis and with the $X$ axis.

1. b) The intervals where the function is positive.

Type: SOMEONE ELSE'S ERROR

In a Mathematics Applied to the Social Sciences exam, the teacher asked to calculate the domain and the intercepts with the axes of the function $f(x) = \frac{x - 3}{\sqrt{2x - 8}}$.

Marcos submitted the following solution:

• Domain: Since there is a square root, I need $2x - 8 \ge 0$, which gives $2x \ge 8 \implies x \ge 4$. Therefore, $\text{Dom}(f) = [4, +\infty)$.

• Intercepts with the axes: For the $Y$-intercept, I set $f(0) = \frac{0 - 3}{\sqrt{0 - 8}} = \frac{-3}{\sqrt{-8}}$, which does not exist, so it does not intersect the $Y$-axis. For the $X$-intercept, I set the numerator equal to zero: $x - 3 = 0 \implies x = 3$, so the intercept point is $(3, 0)$.

Your task:

1. Identify the two conceptual errors made by Marcos (one in the domain and another in the intercepts).

2. Explain in a reasoned way why his deductions are wrong.

3. Provide the corrected solution with the exact domain and the correct conclusion regarding the intercepts with the axes.

Type: INVERSE PROBLEM

A market research consultancy needs to define an elementary rational function of the form:

$$f(x) = \frac{a \cdot x + b}{x + c}$$

to model customer satisfaction level as a function of waiting time $x$ (in minutes).

For the model to be consistent with field observations, the function must simultaneously satisfy these three conditions:

1. Its mathematical domain must be $\mathbb{R} \setminus \{3\}$.

2. It intersects the horizontal axis ($X$) at the point $(-2, 0)$.

3. It intersects the vertical axis ($Y$) at the point $(0, 4)$.

Carry out the following tasks:

1. Deduce step by step and with justification the values of the parameters $a$, $b$, and $c$.

2. Explain why the value obtained for $c$ guarantees that there is only one value excluded from the domain.

3. Once the function is obtained, reason whether it is possible for it to intersect the $X$-axis at any other point different from $(-2, 0)$ without needing to calculate again.

Given the analytical function $f(x) = \frac{\sqrt{x+5}}{x^2 - 16}$, perform the following tasks:

1. Determine the domain of the function, expressing the result in interval notation.

2. Calculate the axis intercepts.

3. Study the sign of the function over its domain.

A seismograph has recorded the intensity of an aftershock following a small earthquake. The graph of the intensity $I$ (in degrees) against time $t$ (in minutes) is given by the following image:

From the graph, identify:

1. The domain and range of the function.

2. The intervals where the intensity is positive.

3. Is there any moment when the intensity is zero? State the values of $t$.

Determine algebraically whether the following functions exhibit any type of symmetry (even, odd, or neither):

1. a) $f(x) = \frac{3x^2}{x^4 + 5}$

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

1. c) $h(x) = 2x^2 - 3x + 1$

A streaming platform offers two monthly plans. The "Premium" plan costs 15 € flat rate per month and includes 100 hours of content. Beyond the 100th hour, the user must pay 0.50 € for each additional hour.

1. Write the algebraic expression of the piecewise function that relates the monthly cost $C$ to the viewing hours $h$.

2. Plot the graph of the function.

3. Determine the range of the function if the maximum number of hours a user can watch per month is 300.

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

The height of a capsule on a Ferris wheel above the ground, in metres, is given by a periodic function. Observe the diagram of a full cycle:

If the Ferris wheel takes 8 minutes to complete one full revolution, and the minimum height is 2 metres and the maximum is 25 metres:

1. State the period of the function.

2. Determine the range of the height function.

3. If we start counting time at the lowest point, what will the height be after 4 minutes? And after 12 minutes?

Study the axis intercepts and the sign of the logarithmic function:

$$f(x) = \log_{2}(x^2 - 3)$$

Note: Remember that to study the sign you must take into account the domain of the function.

Given the function $f(x) = \frac{2x - 6}{x + 3}$, calculate:

1. Its domain.

2. The intercepts with the axes $OX$ and $OY$.

3. Find the value of $x$ for which the image is $y = 1$.

4. Reason whether the value $y = 2$ belongs to the range of the function.

Analyse the following piecewise function:

$$f(x) = \begin{cases} x + 4 & \text{if } x < -2 \\ x^2 - 2 & \text{if } -2 \leq x \leq 2 \\ 2 & \text{if } x > 2 \end{cases}$$

1. Calculate $f(-4)$, $f(0)$ and $f(5)$.

2. Study the continuity at the jump points $x = -2$ and $x = 2$.

3. Determine the intervals where the function has a negative sign.

Type: MODELING

In the bay of Santander, sea level height varies periodically due to the action of the tides. Based on tide gauge records from Puertos del Estado, the water height $h(t)$ in meters, after $t$ hours have elapsed since 00:00 h on a given day, is approximated by the function:

$$h(t) = 3 + 1,8 \cdot \cos\left(\frac{\pi}{6} \cdot t\right), \quad t \in [0, 24]$$

Answer the following questions in a reasoned manner:

1. Justify why this is a periodic function and calculate its period $T$. What physical interpretation does this value have in the daily tidal cycle?

2. Determine the range or image of the function, $\text{Im}(h)$, indicating with justification the maximum height (high tide) and the minimum height (low tide).

3. For pleasure boats to enter the dock safely, a minimum depth of $2,1\text{ m}$ is required. Set up the corresponding inequality and determine, by studying the sign of $h(t) - 2,1$, during which time interval between 00:00 h and 12:00 h it is safe to perform the entry maneuver.

Type: ESTIMATE AND CHECK

The relative economic balance of a Spanish wine cooperative is given by the function $g(x) = \frac{x^2 - 9}{x^2 + 1}$, where $x \ge 0$ represents the volume of processed grapes (in hundreds of tons).

1. Estimate intuitively:

2. If $x = 0$, what sign will the initial balance have?

3. If huge quantities of grapes are processed ($x$ very large), do you think the balance will grow indefinitely towards infinity, or will it tend to stabilize around some specific number? Justify your intuition by looking at the degrees of the numerator and the denominator.

4. Based on the above, write an initial hypothesis about what the approximate range of $g(x)$ will be.

5. Check with rigor:

6. Determine the domain of $g(x)$ if it were defined for all $\mathbb{R}$ and justify whether it has even or odd symmetry.

7. Analytically calculate the intercepts with the axes and indicate the intervals of positive sign ($g(x) > 0$) and negative sign ($g(x) < 0$) for $x \ge 0$.

8. Solve for $x$ from the equality $y = \frac{x^2 - 9}{x^2 + 1}$ to formally calculate the exact range of $g(x)$ for $x \in \mathbb{R}$. Compare this formal result with your prior estimate and explain the meaning of the ceiling or upper bound that you have obtained.

A team of biologists is studying the viability of a population of microorganisms in a controlled environment. The potential growth rate of the colony, as a function of the water temperature $x$ (in degrees Celsius), is modeled by the expression:

$$f(x) = \frac{\sqrt{x^2 - 16}}{\ln(x - 5)}$$

Determine, with reasoning, the domain of definition of this function, taking into account the restrictions of the square root, the natural logarithm and the denominator of the fraction. Interpret the result by stating from what temperature it is possible to start the experiment.

Consider the function $f(x) = \frac{ax^3 + bx}{x^2 + k}$. Find the values of the parameters $a$, $b$ and $k$ knowing that the following conditions are simultaneously satisfied:

1. The function is odd.

2. It has a vertical asymptote at the line $x = 2$.

3. The graph of the function intersects the x-axis at the point $(3, 0)$.

4. It has an oblique asymptote whose gradient is $m = 2$.

Justify algebraically whether, with the values obtained, the function exhibits any additional symmetry with respect to the origin.

The market value of a new cryptocurrency during its first 24 hours after launch has behaved according to the following piecewise function, where $t$ represents the time in hours and $V(t)$ the value in euros:

$$V(t) = \begin{cases} 2^t - 1 & \text{if } 0 \leq t < 4 \\ \dfrac{24t - 36}{t + 2} & \text{if } 4 \leq t \leq 24 \end{cases}$$

1. a) Study the continuity of the function at the break point $t = 4$.

1. b) Determine the range (image) of the function on the whole interval $[0, 24]$.

1. c) Find the sign intervals (when the value is positive) and the points of intersection with the axes.

The mean temperature $T$ (in ºC) of a coastal city over a year can be approximated using a periodic function. It is known that the minimum temperature is $8^\circ\text{C}$ (reached at the beginning of January, $t=0$) and the maximum is $28^\circ\text{C}$ (reached at the beginning of July, $t=6$). The function follows the model:

$$T(t) = A \cdot \cos(Bt) + C$$

1. a) Determine the values of $A$, $B$ and $C$ knowing that the period is 12 months.
2. b) Calculate in which months of the year the temperature is above $23^\circ\text{C}$ (intervals of sign for the difference $T(t) - 23$).
3. c) State the points of intersection of the function with the horizontal line $y = 18$.

Let the function be $f(x) = |x^2 - 2x - 3|$.

1. a) Express the function as a piecewise-defined function by removing the absolute value.

1. b) Find the points of intersection with the axes and study the sign of the original function.

1. c) Determine the domain and range of the function. Is the function symmetric with respect to the line $x = 1$? Justify your answer.

Given the rational function $g(x) = \frac{2x^2 - 8}{x^2 - 5x + 6}$:

1. a) Determine its domain. Pay special attention to the points where the numerator is also zero.
2. b) Identify the asymptotes (vertical, horizontal or oblique) and study the position of the curve relative to them.
3. c) Find the range of the function by analysing whether there is any value of $y$ that cannot be reached.
4. d) Study the intervals where the function is positive and where it is negative.

A financial analyst uses the function $P(x) = \log_2(x^2 - kx + 9)$ to predict the behaviour of a stock market index.

1. a) For which values of the parameter $k$ is the domain of the function the set of all real numbers $\mathbb{R}$?
2. b) If $k = 6$, determine the domain of the function and the x-axis intercepts.
3. c) For $k = 0$, study the symmetry of the function and explain whether it is even, odd or neither.

Sketch the graph of a function $f(x)$ that simultaneously satisfies all of these properties:

1. Its domain is $\mathbb{R} - \{-2, 2\}$.

2. It is an even function.

3. It has vertical asymptotes at $x = -2$ and $x = 2$, such that $\lim_{x \to 2^+} f(x) = +\infty$ and $\lim_{x \to 2^-} f(x) = +\infty$.

4. It has a horizontal asymptote at $y = 1$ as $x \to \infty$.

5. The intersection points with the OX axis are $(-1, 0)$ and $(1, 0)$, and it does not cross the OX axis at any other point.

6. The intersection point with the OY axis is $(0, -2)$.

Once you have sketched it, state, with justification, the intervals where the function has each sign.

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

Type: CHOOSE THE METHOD

To analyze a complex economic function, two students must study the symmetry and sign of the function:

$$f(x) = \frac{\ln(x^2 - 3)}{x}$$

• Álvaro's method (analytical brute force): He calculates $f(-x)$ algebraically by directly substituting the variable. Then, to study the sign, he finds all roots of the numerator by solving $\ln(x^2 - 3) = 0$, those of the denominator $x = 0$, and the existence boundaries of the root/logarithm, constructing a table with 6 intervals on the real line.

• Lucía's method (structural properties and decomposition): She first observes that the numerator $N(x) = \ln(x^2 - 3)$ is an even function and the denominator $D(x) = x$ is an odd function. From the quotient rule for symmetric functions, she immediately deduces the overall type of symmetry of $f(x)$. She argues that, thanks to this symmetry, it is only necessary to study the intercepts and sign on the positive half-line of the domain and then reflect the conclusions onto the negative half-line.

Answer the following parts:

1. Rigorously determine the domain of $f(x)$. Why is the point $x = 0$ outside the function even before checking that it makes the denominator zero?

2. Evaluate the mathematical validity of Lucía's method: formally prove that the quotient of an even function by an odd function results in an odd function ($f(-x) = -f(x)$).

3. Justify which of the two methods minimizes algebraic work and reduces the probability of making sign errors. Apply the most efficient method to determine the intervals where $f(x) > 0$ and where $f(x) < 0$.

Type: MODELING

An express courier company in Spain wants to update its pricing structure for heavy packages. The billing rate $C(x)$ (in euros) as a function of the package mass $x$ (in kilograms, with $x > 0$) must follow the following piecewise model:

$$C(x) = \begin{cases} a + \sqrt{b \cdot x} & \text{if } 0 < x \le 4 \\[2mm] \dfrac{k \cdot x}{x - 2} & \text{if } x > 4 \end{cases}$$

The finance department imposes three mandatory conditions:

• Sending a $1\text{ kg}$ package costs $7\text{ €}$.

• Sending a $4\text{ kg}$ package costs $10\text{ €}$.

• The rate must not have abrupt jumps: the cost value must coincide at the boundary of both intervals ($x = 4$).

Carry out the following analysis and design tasks:

1. Determine with justification the values of the constants $a$, $b$, and $k$.

2. Analyze the domain of the function in its real context. In the second branch there is a denominator that vanishes at $x = 2$; reason whether that value generates a discontinuity or asymptote in the rate function $C(x)$ or not.

3. Study the range of $C(x)$ for all $x > 0$. Based on that range, argue whether a customer could ever pay a bill of $20\text{ €}$ for an extremely heavy package, or if there is an upper limit cost that will never be exceeded.

4. Does it make mathematical sense to study whether this function is even or odd? Justify your answer considering exclusively the formal definition of symmetry and the domain of definition of $C(x)$.