Consider the following functions

Question

f(x)=1/ sqrt(x) , g(x)=x²-2x

Find (f ∘ g)(x).

 

Find the domain of (f ∘ g)(x).

 (Enter your answer using interval notation.)

 

Find (g ∘ f)(x).

 

Find the domain of (g ∘ f)(x).

 (Enter your answer using interval notation.)

 

Find (f ∘ f)(x).

 

Find the domain of (f ∘ f)(x).

 (Enter your answer using interval notation.)

 

Find (g ∘ g)(x).

 

Find the domain of (g ∘ g)(x).

 (Enter your answer using interval notation.)

 

Answer 1

For f(gx) we substitute g(x) for x in f(x)

which gives us 1/sq root of x^2-2x. Since you can’t take the sq root of a negative number the denominator has to be >0. This occurs when x>2 or when x is negative ( in the later case when x is negative both x^2 and -2x are both +. In interval notation this is 0>x or >2

For g(fx) we substitute f(x) for all x’s in g(x) which gives us ( 1/sq root of x)^2 -2/sq root of x. Since we have a square root in the equation the domain will be limited to x>0

For f(fx) we get 1/sqroot of the sqroot of x which is 1/4throot of x. This also will lie it the domain to x>0

If you want to check these graph and inspect these equations in desmos