Consider the following functions.

f(x) = x2 − 3x, g(x) =√(x+2)
(a) Find (f *g)(x). (Simplify your answer completely.)
(B)Find the domain of f *g. (Enter your answer using interval notation.)
(c) Find (g *f)(x). (Simplify your answer completely.)
(D) Find the domain of g*f. (Enter your answer using interval notation.)

f*g = g*f = (x^2-3x)√(x+2)

domain is all reals x >= -2

If you meant ◦ instead of *, then

(f◦g) = f(g) = g^2-3g = (x+2) - 3√(x+2)
(g◦f) = g(f) = √(f+2) = √(x^2-3x)

domain of f◦g is x >= -2
domain of g◦f is x<= 0 or x>= 3

To find the composite function (f * g)(x), we substitute the expression for g(x) into f(x):

(f * g)(x) = f(g(x)) = f(√(x + 2))

(a) Simplifying the expression completely:

f(x) = x^2 - 3x
g(x) = √(x + 2)

Substituting g(x) into f(x):

(f * g)(x) = (√(x + 2))^2 - 3(√(x + 2))

Simplifying further:

(f * g)(x) = (x + 2) - 3√(x + 2)
(f * g)(x) = x + 2 - 3√(x + 2)

Therefore, (f * g)(x) = x + 2 - 3√(x + 2).

(b) To find the domain of f * g, we need to consider the domain of both f(x) and g(x).

The domain of f(x) is all real numbers since there are no restrictions on x.

The domain of g(x) is determined by the square root function, which requires the expression inside the square root to be non-negative:

x + 2 ≥ 0
x ≥ -2

Therefore, the domain of g(x) is x ≥ -2.

Combining the domains of f(x) and g(x), we have:

Domain of f * g: x ≥ -2

Therefore, the domain of f * g in interval notation is [ -2, ∞ ).

(c) Now, let's find the composition (g * f)(x). This means we substitute the expression for f(x) into g(x):

(g * f)(x) = g(f(x)) = g(x^2 - 3x)

(c) Simplifying the expression completely:

g(x) = √(x + 2)
f(x) = x^2 - 3x

Substituting f(x) into g(x):

(g * f)(x) = √((x^2 - 3x) + 2)

(g * f)(x) = √(x^2 - 3x + 2)

Therefore, (g * f)(x) = √(x^2 - 3x + 2).

(d) To find the domain of g * f, we need to consider the domain of both g(x) and f(x).

The domain of g(x) is x ≥ -2 (as determined in part b).

The domain of f(x) is all real numbers since there are no restrictions on x.

Combining the domains of g(x) and f(x), we have:

Domain of g * f: x ≥ -2

Therefore, the domain of g * f in interval notation is [ -2, ∞ ).

To find the compositions (f * g)(x) and (g * f)(x), we need to substitute one function into another.

(a) To find (f * g)(x):
(f * g)(x) = f(g(x))
Substitute g(x) into f(x):
(f * g)(x) = f(√(x+2))
Replace f(x) in this expression with its definition:
(f * g)(x) = (√(x+2))^2 - 3(√(x+2))
Simplify the expression:
(f * g)(x) = x + 2 - 3√(x+2)

(b) To find the domain of f * g, we need to determine which values of x are valid inputs for the resulting expression. Since the square root (√) is defined only for non-negative numbers, the expression √(x+2) is valid only if x+2 ≥ 0. Solving this inequality:
x + 2 ≥ 0
x ≥ -2
The domain of f * g is x ≥ -2, or in interval notation: [-2, ∞).

(c) To find (g * f)(x):
(g * f)(x) = g(f(x))
Substitute f(x) into g(x):
(g * f)(x) = g(x^2 - 3x)
Replace g(x) in this expression with its definition:
(g * f)(x) = √((x^2 - 3x) + 2)
Simplify the expression if possible.

(d) To find the domain of g * f, we need to determine which values of x are valid inputs for the resulting expression. The square root (√) is defined only for non-negative numbers, so the expression (x^2 - 3x) + 2 inside the square root must be greater than or equal to zero. Solving this inequality:
x^2 - 3x + 2 ≥ 0
Factorizing the left side:
(x - 1)(x - 2) ≥ 0
The solutions of this inequality are 1 and 2 (including them), and the intervals where the expression is positive or zero are (-∞, 1] and [2, ∞). Therefore, the domain of g * f is x ≤ 1 or x ≥ 2, or in interval notation: (-∞, 1] ∪ [2, ∞).