Question

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 1

Answer:

Given functions,

$f(x) = 3 - x^2$

$g(x) = \sqrt{x}+1$

Since, by the compositions of functions,

1. (g◦f)(x) = g(f(x))

$=g(3-x^2)$

$=\sqrt{3-x^2}+1$

Since, (g◦f) is defined,

If 3 - x² ≥ 0

⇒ 3 ≥ x²

⇒ -√3 ≤ x ≤ √3

Thus, Domain = [-√3, √3]

2. (f◦g)(x) = f(g(x))

$=f(\sqrt{x}+1)$

$=3-(\sqrt{x}+1)^2$

Since, (g◦f) is defined,

If  x ≥ 0

Thus, Domain = [0, ∞)

3. (f◦f)(x) = f(f(x))

$=f(3-x^2)$

$=3-(3-x^2)^2$

$=3-9-x^4+6x^2$

$=-6+6x^2-x^4$

Since, (f◦f) is a polynomial,

We know that,

A polynomial is defined for all real value of x,

Thus, Domain = (-∞, ∞)