domain of sqrt(3-x)+sqrt(x^2-4) — step-by-step solution

1) Let's evaluate the domain of

\sqrt{3 - x} + \sqrt{x^{2} - 4}

For the expression

\sqrt{3 - x}

to exist, you must check:

3 - x \geq 0

For the expression

\sqrt{x^{2} - 4}

to be defined, we must have:

x^{2} - 4 \geq 0

2) Let's solve in

ℝ

3 - x \geq 0

ⓘ We add

x

to both sides

3 \geq x

Solution obtained:

x \leq 3

3) Let's solve in

ℝ

x^{2} - 4 \geq 0

An even power only depends on the absolute value:

x^{2} \geq 4

is equivalent to

| x | \geq 2

| A | \geq B

means that:

A \geq B

A \leq - B

Solution found:

x \in (-∞, -2] \cup [2, +∞)

Let's create the table of the different solutions to find the overall solution:

▶Domain of definition:

x \in (-∞, -2] \cup [2, 3]