Find the rules for (f∘g)(x) and (g∘f)(x) and give the domain of each composite function.

f(x) = x^2 g(x) = √(1-x)
The Domain of (f∘g)(x) is :
(f∘g)(x) =
The Domain of (g∘f)(x) is :
(g∘f)(x) =

(f∘g)(x) = [ √(1-x) ]^2 = 1-x all real x

(g∘f)(x) = √(1-x^2) which is imaginary for |x| >1 so only from x = -1 to x =+1

wait that is wrong

To find the rules for the composite functions (f∘g)(x) and (g∘f)(x), we need to substitute g(x) into f(x) and f(x) into g(x) respectively.

1. (f∘g)(x):
To find (f∘g)(x), we substitute g(x) into f(x):
(f∘g)(x) = f(g(x)) = f(√(1-x)) = (√(1-x))^2 = (1-x)

The rule for (f∘g)(x) is: (f∘g)(x) = 1 - x

Regarding the domain of (f∘g)(x), the function (f∘g)(x) is defined for all real numbers since there are no restrictions on the input x.

2. (g∘f)(x):
To find (g∘f)(x), we substitute f(x) into g(x):
(g∘f)(x) = g(f(x)) = g(x^2) = √(1 - x^2)

The rule for (g∘f)(x) is: (g∘f)(x) = √(1 - x^2)

Regarding the domain of (g∘f)(x), we know that the input for g(x) must be a non-negative number since it involves taking the square root. Therefore, the domain of (g∘f)(x) is all x values such that 1 - x^2 ≥ 0. Solving this inequality, we obtain -1 ≤ x ≤ 1.

In summary:
- The rule for (f∘g)(x) is: (f∘g)(x) = 1 - x, and its domain is all real numbers.
- The rule for (g∘f)(x) is: (g∘f)(x) = √(1 - x^2), and its domain is -1 ≤ x ≤ 1.

To find the rules for (f∘g)(x) and (g∘f)(x), we need to first find the composition of the two functions.

1. (f∘g)(x):
To find (f∘g)(x), we substitute g(x) into f(x):
(f∘g)(x) = f(g(x)) = f(√(1-x)) = (√(1-x))^2 = 1 - x.

Therefore, the rule for (f∘g)(x) is (1 - x).

2. (g∘f)(x):
To find (g∘f)(x), we substitute f(x) into g(x):
(g∘f)(x) = g(f(x)) = g(x^2) = √(1 - x^2).

Therefore, the rule for (g∘f)(x) is √(1 - x^2).

Now let's determine the domain of each composite function.

1. The Domain of (f∘g)(x):
The function (f∘g)(x) is defined as 1 - x. To find the domain, we need to consider any restrictions on x. In this case, the only restriction is the square root in the function g(x), which requires the argument (1 - x) to be greater than or equal to zero.
So, we have the inequality: 1 - x ≥ 0.
Solving for x, we get x ≤ 1.

Therefore, the domain of (f∘g)(x) is x ≤ 1.

2. The Domain of (g∘f)(x):
The function (g∘f)(x) is defined as √(1 - x^2). To find the domain, we need to consider any restrictions on x. In this case, the square root requires the argument (1 - x^2) to be greater than or equal to zero.
So, we have the inequality: 1 - x^2 ≥ 0.
Solving for x, we get -1 ≤ x ≤ 1.

Therefore, the domain of (g∘f)(x) is -1 ≤ x ≤ 1.