Domain, Range, Symmetry, Periodicity, Intercepts, and Sign

Determine the domain of the following analytical functions:

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

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

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

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

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.

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.

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?

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.

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

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

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.

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?

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).

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.

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 ($f(x) < 0$).

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.

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.

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 $t$.

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.