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Tomtit [17]
2 years ago
8

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)
Mathematics
1 answer:
Tamiku [17]2 years ago
8 0

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 = (-∞, ∞)

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