Question

Find (f∘g)(x) when f(x) = √(x + 3)/x and g(x) = √(x + 3)/(2x).

a) (f∘g)(x) = √((x + 3)/(2x))/x

b) (f∘g)(x) = √((x + 3)/(x))/(2x)

c) (f∘g)(x) = (x + 3)/(2√(x^2))

d) (f∘g)(x) = (2x√(x + 3))/(x)

Answer 1

To find (f ∘ g)(x) when f(x) and g(x) are given, we substitute g(x) into f(x). Further algebraic manipulation would be needed than what is reflected in the offered answers, which suggests a potential error in the provided options or in the understanding of the functions.

Step-by-step explanation:

To find (f ∘ g)(x) when f(x) = √(x + 3)/x and g(x) = √(x + 3)/(2x), we need to substitute g(x) into f(x). This means we will replace every instance of x in f(x) with g(x):

f(g(x)) = f(√(x + 3)/(2x)) = √((√(x + 3)/(2x)) + 3)/(√(x + 3)/(2x))

However, before we calculate this new expression, remember that the square root of a square brings us back to the original number (x² = √x) and that fractional exponents indicated a root (such as x to the power of 1/2 is the square root of x). This turns the problem into:

f(g(x)) = (√(x + 3) + 3x/(2x))/(√(x + 3)/(2x))

But it is evident that both the numerator and denominator will not simplify into a square root over x. So none of the offered answers conform to the calculations required for f(g(x)). Therefore, to find the correct expression for (f ∘ g)(x), further algebraic manipulation would be needed, which is not reflected in the options given.