find possible functions

Question

If h(x) = 4+1/(sq root of x-5), find possible functions f and g so that h(x)= (fog)(x). DO NOT DO NOT DO NOT use f(x)=x or g(x)=x. Check solution

Answer 1

Let g(x) = √(x-5), f(x) = 4 + 1/x

 

Then (fοg)(x) = f(g(x)) = f(√(x-5)) = 4 + 1/√(x-5) = h(x)

Answer 2

Let h(x) = 4+1/sqrt(5) = (4+1/sqrt(5))*sqrt(x-5)/sqrt(x-5)=4+sqrt(x-5)/(x-5) == 4+g(x)/g(x)^2.

f(x)=4+x/x^2, g(x)=sqrt(x+5). f(g(x)= 4+sqrt(x+5)/(sqrt(x+5))^2=4+1/sqrt(x+5)=h(x).