Intervals (revision)

In electronic component design, the voltage tolerance of a sensor is defined by the set of real values $x$ satisfying the following inequality:

$-3 \leq x < 4,5$

Express this set of values using interval notation and state the type of interval (open, closed, or half-open).

A tech company's data server maintains its optimal operating temperature in the interval given by $I = (-10, 35]$. Represent this interval on the number line and determine whether the following temperature values belong to the set:

• $a = -10$

• $b = 0$

• $c = 35$

• $d = 35,1$

Find the intersection and the union of the following sets of real numbers. Express the final result using interval notation:

$A = \{x \in \mathbb{R} : -2 \leq x < 5\}$

$B = \{x \in \mathbb{R} : |x - 3| \leq 4\}$

Given the intervals $I = (-\infty, 4]$ and $J = (-2, +\infty)$, perform the following operations and express the result in interval notation and graphically:

1. a) $I \cap J$
2. b) $I \cup J$
3. c) $\mathbb{R} - (I \cap J)$

Let sets $S_1$ and $S_2$ be defined by the following conditions. Calculate exactly the set resulting from the operation $(S_1 \cup S_2)^c$ (complement of the union):

Given the rational function $f(x) = \frac{1}{x^2 - 9}$, determine its domain and express it as a union of intervals. Then, state whether the value $x = 3,0001$ belongs to a neighbourhood with centre $3$ and radius $\delta = 0,001$.

In the design of a precision part for a telescope, the radius of curvature $R$ must belong to the domain of the following function to ensure structural stability. Find said domain and express it as a union of intervals:

$f(R) = \sqrt{\frac{\ln(R^2 - 8)}{R - 4}}$

Consider the function $g(x) = \log_2\left( |x^2 - 1| - 2 \right)$.

1. a) Determine the set of real values of $x$ for which the function is not defined.

1. b) Express this set as an interval or as a union of intervals.

1. c) Classify each interval obtained as open, closed or half-open.

In the study of limits, the concept of a deleted neighbourhood is essential. If we have the neighbourhood $E(3; 0,5)$, which represents all real numbers whose distance to the centre $3$ is less than $0,5$:

1. a) Write this set in interval form.
2. b) Write the deleted neighbourhood $E^*(3; 0,5)$, that is, the same set but excluding the central point.

Find the set of solutions of the following inequality with nested moduli. Represent the solution on the number line and specify whether it is a bounded set, stating its supremum and infimum if so:

Determine the value of the real parameter $k > 0$ so that the intersection of the sets $A$ and $B$ is an interval of length exactly equal to $3$ units, where:

Hint: express $A$ and $B$ as intervals of the real line and study, depending on the value of $k$, what $A \cap B$ looks like.