Given f(x) = (1)/(x-5) and g(x)= √(x-2) Find (f o g) (x) and write the domain of (fog) (x) in interval form. can someone solve this step by step

Question

Given f(x) =
`(1)/(x-5)` and g(x)=
`√(x-2)` Find (f o g) (x) and write the domain of (fog) (x) in interval form. can someone solve this step by step

Answer 1

`f(g(x)) =(1)/(√(x - 2) - 5)`

`D(f(g(x)) = (2;27) \cup (27; +\infty)`

Explanation:

`f(x) = (1)/(x - 5)`

`g(x) = √(x - 2)`

`f(g(x)) = (1)/(g(x) - 5) = (1)/(√(x - 2) - 5)`

`D(f(g(x)):`

`\displaystyle \left \{ {{x - 2 > 0} \atop { √(x - 2) - 5 \\eq 0 }} \right.`
`\displaystyle \left \{ {{x > 2} \atop { √(x - 2) \\eq 5 }} \right`
`\displaystyle \left \{ {{x > 2} \atop { (√(x - 2))^(2) \\eq 5^(2) }} \right`
`\displaystyle \left \{ {{x > 2} \atop { x - 2 \\eq 25 }} \right`
`\displaystyle \left \{ {{x > 2} \atop { x \\eq 27 }} \right`

`D(f(g(x)) = (2;27) \cup (27; +\infty)`