Domain, Symmetry, Intercepts, and Monotonicity

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) $h(x) = \frac{1}{x^2 - 9}$

A group of 2nd-year Bachillerato students has designed an app for organizing events. The expected profit, in pesos, 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.

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.

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

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."

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.

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.

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?