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 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.

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

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?

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."