To find the formulas for the compositions f ◦ g and g ◦ f, we need to substitute the expressions for f(x) and g(x) into the composition formulas.
1. f ◦ g:
The composition f ◦ g means we apply g(x) first and then apply f(x) to the result. Therefore, f ◦ g(x) can be calculated as:
f ◦ g(x) = f(g(x))
Substituting the given functions:
f(g(x)) = f(√(1 - x))
Using the formula for f(x) = x^2:
f(g(x)) = (√(1 - x))^2
f(g(x)) = 1 - x
So, the formula for f ◦ g is f(g(x)) = 1 - x.
2. g ◦ f:
The composition g ◦ f means we apply f(x) first and then apply g(x) to the result. Therefore, g ◦ f(x) can be calculated as:
g ◦ f(x) = g(f(x))
Substituting the given functions:
g(f(x)) = g(x^2)
Using the formula for g(x) = √(1 - x):
g(f(x)) = √(1 - (x^2))
So, the formula for g ◦ f is g(f(x)) = √(1 - x^2).
As for the domains of the compositions:
- For f ◦ g(x) = 1 - x, the domain is all real numbers since there are no restrictions on the input.
- For g ◦ f(x) = √(1 - x^2), the domain is -1 ≤ x ≤ 1. This is because the expression inside the square root (√(1 - x^2)) must be non-negative, so 1 - x^2 ≥ 0. Solving this inequality gives -1 ≤ x ≤ 1.
Therefore, the domain of f ◦ g is all real numbers, and the domain of g ◦ f is -1 ≤ x ≤ 1.