Exercises: Domain, symmetry, axes intercepts and monotonicity

A group of 2nd-year Bachillerato students has designed an app for organizing events. The expected profit, in euros, depends on the number of registered users $x$ (in hundreds) according to the function:

$B(x) = x^2 - 10x + 8$

You are asked to:

1. a) Determine the domain of the function, taking into account that $x$ represents a number of users (it cannot be negative).

1. b) Calculate the intercepts of the graph with the coordinate axes.

Study the symmetry of the following functions, stating in each case whether they are even, odd, or have no symmetry:

1. $f(x) = x^4 - 5x^2 + 2$

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

Study the intervals of increase and decrease (monotonicity) of the polynomial function:

$f(x) = -x^2 + 8x - 12$

1. a) State the coordinates of the vertex and specify whether it is a relative maximum or minimum.

1. b) Determine the intervals where $f$ is increasing and decreasing.

Find the domain of definition of each of these functions, justifying your answer:

• a) $f(x) = \sqrt{x - 3}$

• b) $g(x) = \ln(x + 2)$

• c) $h(x) = \frac{1}{x^2 - 9}$

Lucas has to determine the domain and the axis intercepts of the function $f(x) = \dfrac{x^2 - 4}{x - 2}$. In his notebook, he writes the following resolution:

"Since the numerator factors as $(x - 2)(x + 2)$, I simplify the fraction by dividing by $(x - 2)$ and get $f(x) = x + 2$. From here I conclude:

1. The domain is $\mathbb{R}$, because a first-degree polynomial function is defined for any real number.

2. With the $Y$-axis, it intercepts at $(0, 2)$ when calculating $f(0)$. With the $X$-axis, I solve $x^2 - 4 = 0$, which gives $x = 2$ and $x = -2$, so there are two intercepts: $(2, 0)$ and $(-2, 0)$."

Identify the two conceptual errors that Lucas made, explain with mathematical precision why his reasoning is flawed, and write the correct solution for both parts (domain and intercepts).

Consider the polynomial function $f(x) = -x^3 + 12x$.

1. a) Before computing any derivative or solving equations:

1. Observing the exponents of its terms, estimate what symmetry its graph will display and what immediate consequence this symmetry has on the $Y$-axis intercept.

2. Reason how many $X$-axis intercepts at most you expect to find.

3. For large positive values of $x$ (for example, $x = 100$), estimate the sign that $f(x)$ will have and what this suggests about whether the function will end up increasing or decreasing to the right.

1. b) Carry out the analytical calculations: formally prove the symmetry by checking $f(-x)$, find all the axis intercepts, and determine the intervals of increase and decrease using the sign of $f'(x)$.
2. c) Compare the obtained results with your estimates from part a) and argue whether your prior estimation was accurate.

Consider the function $f(x) = \sqrt{4 - x^2}$.

• a) Determine its domain of definition.

• b) Find the axes intercepts.

• c) Study its symmetry and, with the above information, sketch its graph. What familiar shape does it form?

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

The number of members of a book club during its initial years is given by $N(t) = t^3 - 12t + 60$, where $t$ is in years and $0 \leq t \leq 4$.

• a) Study in which time intervals the club gained and lost members.

• b) Find the relative extrema and interpret them in context.

• c) What was the minimum number of members during this period?

According to operation reports from Red Eléctrica de España, on a clear spring day the peninsular photovoltaic power $P(t)$, measured in gigawatts (GW), can be approximated between $06:00\text{ h}$ ($t = 6$) and $20:00\text{ h}$ ($t = 20$) by the quadratic function:

$P(t) = -0,1 \cdot t^2 + 2,6 \cdot t - 12$.

1. a) Justify what the contextual domain of the function is and check whether the estimated power makes physical sense at the endpoints of this time interval.
2. b) Find the intercepts of $P(t)$ with the horizontal axis of abscissas and interpret their meaning regarding the effective sunrise and sunset of solar generation.
3. c) Using the first derivative $P'(t)$, calculate at what exact solar time the maximum energy production is recorded and determine what that peak power in GW is.
4. d) Determine the time intervals in which electricity generation is increasing and those in which it is decreasing. Furthermore, justify whether the generation curve has symmetry with respect to the vertical line $t = 13$ by checking if $P(13 - h) = P(13 + h)$ holds for any time offset $h$.

A digital marketing consultancy needs to design a model for the conversion rate $f(x)$ of a website as a function of the time spent on the page $x$ (in minutes). The model follows the rational structure $f(x) = \dfrac{a \cdot x}{x^2 + b}$.

Determine the values of the real parameters $a$ and $b$ so that the following three conditions are simultaneously met:

1. The domain of the function is all of $\mathbb{R}$ (the denominator does not vanish for any real value).

2. The curve passes through the point $(2, 1)$.

3. The function has a relative extremum at the abscissa $x = 2$.

Once the values of $a$ and $b$ are fixed:

1. a) Analytically prove the symmetry of the function (even, odd, or neither) and interpret what it geometrically reflects on the graph.
2. b) Find all the intercepts with the coordinate axes.
3. c) By studying the sign of the derivative $f'(x)$ around $x = 2$, formally justify whether there is a relative maximum or minimum at that point.

Determine the values of the parameter $a$ for which the domain of the function $f(x) = \frac{1}{x^2 + ax + 4}$ is all of $\mathbb{R}$. Reason your answer using the discriminant of the denominator.

State with reasons whether each statement is true or false:

• a) "If $f$ is an odd function defined at $x = 0$, then necessarily $f(0) = 0$."

• b) "If a function is even, its graph cannot cross the $X$-axis."

• c) "The sum of two odd functions is always an odd function."

Two 2nd year Bachillerato students discuss how to study the domain, monotonicity, and relative extrema of the logarithmic function $f(x) = \ln(x^2 + 2x + 5)$:

• Marta's method: Compute $f'(x) = \dfrac{u'(x)}{u(x)}$, solve the equation $f'(x) = 0$, place the roots on the real line, and evaluate the sign of the derivative in each interval.

• Sergio's method: First analyze the properties of the inner parabola $u(x) = x^2 + 2x + 5$ (concavity and coordinates of its vertex) and use the property that the natural logarithm function $y = \ln(u)$ is strictly monotonically increasing on its entire domain $(0, +\infty)$ to transfer the behavior of $u(x)$ to $f(x)$ without differentiating the logarithm.

1. a) Reasonably compare both methods: indicate whether both require checking the domain of $f(x)$ beforehand, argue which of the two strategies requires less algebraic effort, and which one carries a lower risk of sign or differentiation errors.
2. b) Apply Sergio's method (or Marta's, justifying your choice) to:

1. Determine the domain of definition of $f(x)$.

2. Obtain the intervals of increase, decrease, and the exact coordinates $(x, y)$ of its relative extrema.

3. Reason whether the function $f(x)$ possesses even or odd symmetry with respect to the axes.

A freight transport company between Madrid and Zaragoza models the fuel and maintenance cost per trip, $C(v)$ in euros, as a function of the average speed $v$ (in $\text{km/h}$), using the function:

$C(v) = \dfrac{k}{v} + c \cdot v$, defined for $v > 0$.

1. a) Explain the difference between the mathematical domain of $C(v)$ and the domain with physical and regulatory sense on a Spanish motorway (taking into account the legal minimum and maximum speeds: $60\text{ km/h}$ and $120\text{ km/h}$).
2. b) Telemetry data indicates that at an average speed of $100\text{ km/h}$ the cost is exactly $65$ euros per trip and that at this speed the operating cost is minimal. Set up a system and calculate the exact values of the constants $k$ and $c$.
3. c) From the first derivative $C'(v)$, determine the intervals of increase and decrease of the function on the legal domain $[60, 120]$.
4. d) Reason whether it makes sense to study the even or odd symmetry of $C(v)$ both from an analytical point of view and from the driving context.
5. e) Due to roadworks, the Directorate-General for Traffic temporarily limits the maximum speed along the entire section to $90\text{ km/h}$. Based exclusively on the monotonicity deduced in part c), justify whether driving at the maximum speed permitted in roadworks will reduce or increase the cost of the trip compared to the optimal speed of $100\text{ km/h}$, without calculating the numerical value of $C(90)$.