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

A smart device records the battery level of a cell 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 pesos and $B(x)$ is the profit in hundreds of pesos).

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 liters 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.

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 pesos flat rate per month and includes 100 hours of content. Beyond the 100th hour, the user must pay 0.50 pesos 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 meters, 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 meters and the maximum is 25 meters:

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.

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 pesos:

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

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:

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

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 behavior 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.