domain of f(x)=(sqrt(x-2))/(2x-10)

We save the definition:

f (x) = \frac{\sqrt{x - 2}}{2 x - 10}

1) Let's find the domain of

\frac{\sqrt{x - 2}}{2 x - 10}

For the expression

\frac{\sqrt{x - 2}}{2 x - 10}

to be defined, we must have:

2 x - 10 \neq 0

For the expression

\sqrt{x - 2}

to exist, you must check:

x - 2 \geq 0

2) Let's solve in

ℝ

2 x - 10 \neq 0

ⓘ We add

10

to both sides

2 x \neq 10

ⓘ Both sides are divided by

2

x \neq \frac{10}{2}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{10}{2}

5

Solution found:

x \neq 5

3) Let's solve in

ℝ

x - 2 \geq 0

ⓘ We add

2

to both sides

Solution found:

x \geq 2

Let's aggregate the different solutions to find the overall solution:

▶Domain of definition:

x \in [2, 5) \cup (5, +∞)