Question

Find the domain of f space space left parenthesis g space left parenthesis x right parenthesis right parenthesis space i f space space f space left parenthesis x right parenthesis space equals space fraction numerator 3 over denominator x space minus space 5 end fraction space space a n d space space g space left parenthesis x right parenthesis space equals space square root of x space minus space 2 end root. Find the domain of f(g(x)) if f(x) = 3/x-5 to and g(x)=√x-2

Answer 1

To find the domain of f(g(x)), the radicand of g(x) must be greater than or equal to 0, and x cannot be equal to 5. Therefore, the domain of f(g(x)) is all values of x greater than or equal to 2, except x=5.

Step-by-step explanation:

To find the domain of f(g(x)), we need to determine the values of x that we can plug into g(x) and then into f(x) without causing any undefined or imaginary outputs.

First, let's find the domain of g(x). Since g(x) involves the square root of x, the radicand (x-2) must be greater than or equal to 0, which means x-2 >= 0. Solving this inequality, we find x >= 2.

Next, let's find the domain of f(g(x)). Since f(x) involves dividing by (x-5), we need to make sure that x-5 is not equal to zero. Therefore, x cannot be equal to 5. So, the domain of f(g(x)) is all values of x greater than or equal to 2, except x=5.