Use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation.(g◦f)(x) (f ◦g)(x) (f ◦f)(x)f(x)= 3−x2, g(x)= (√x+1)

Answer:Given functions,[tex]f(x) = 3 - x^2[/tex][tex]g(x) = \sqrt{x}+1[/tex]Since, by the compositions of functions,1. (g◦f)(x) = g(f(x))[tex]=g(3-x^2)[/tex][tex]=\sqrt{3-x^2}+1[/tex]Since, (g◦f) is defined,If 3 - x² ≥ 0⇒ 3 ≥ x² ⇒ -√3 ≤ x ≤ √3Thus, Domain = [-√3, √3]2. (f◦g)(x) = f(g(x))[tex]=f(\sqrt{x}+1)[/tex][tex]=3-(\sqrt{x}+1)^2[/tex]Since, (g◦f) is defined,If  x ≥ 0Thus, Domain = [0, ∞)3. (f◦f)(x) = f(f(x))[tex]=f(3-x^2)[/tex][tex]=3-(3-x^2)^2[/tex][tex]=3-9-x^4+6x^2[/tex][tex]=-6+6x^2-x^4[/tex]Since, (f◦f) is a polynomial,We know that,A polynomial is defined for all real value of x,Thus, Domain = (-∞, ∞)