Find the functions a) f o g, b) g o f, c) f o f, d) g o g, and their domains.
f(x)= square root of x, g(x)= cube root of 1-x
These are my answers, but I am not sure about them and I only figured out one domain... that is the part that really confuses me. Help please!
a) 5th root of 1-x
Domain?
b) The cube root of (1 minus the square root of x)
Domain?
c) Fourth root of x
Domain= [0, infinity]
d) The cube root of (1 minus the cube root of 1 minus x)
Domain?
To find the composite functions f o g, g o f, f o f, and g o g, we need to substitute the expression of one function into the other and evaluate it. Let's go step by step:
a) f o g: Substitute g(x) into f(x)
f o g(x) = f(g(x))
f o g(x) = f(cube root of (1-x))
f o g(x) = square root of (cube root of (1-x))
To determine the domain of f o g(x), we need to consider the domains of both f(x) and g(x). The domain of g(x) is all real numbers since there are no restrictions on input values. However, the domain of f(x) is non-negative real numbers or [0, infinity) because the square root is only defined for non-negative numbers. Therefore, the domain of f o g(x) is the set of values that make (1-x) non-negative, which is x <= 1.
b) g o f: Substitute f(x) into g(x)
g o f(x) = g(f(x))
g o f(x) = g(square root of x)
g o f(x) = cube root of (1 - square root of x)
To determine the domain of g o f(x), we need to consider the domains of both g(x) and f(x). Again, the domain of g(x) is all real numbers. The domain of f(x) is x >= 0 since the square root is only defined for non-negative numbers. Therefore, the domain of g o f(x) is the set of values that make (1 - square root of x) non-negative, which is x <= 1.
c) f o f: Substitute f(x) into f(x)
f o f(x) = f(f(x))
f o f(x) = f(square root of x)
f o f(x) = square root of (square root of x)
f o f(x) = (x)^(1/4)
The domain of f o f(x) is the set of non-negative real numbers, [0, infinity), since the square root is only defined for non-negative numbers.
d) g o g: Substitute g(x) into g(x)
g o g(x) = g(g(x))
g o g(x) = g(cube root of (1-x))
g o g(x) = cube root of (1 - cube root of (1-x))
To determine the domain of g o g(x), we again need to consider the domain of g(x). The domain of g(x) is all real numbers. Therefore, the domain of g o g(x) is the set of all real numbers.
In summary:
a) f o g(x) = square root of (cube root of (1-x))
Domain = x <= 1
b) g o f(x) = cube root of (1 - square root of x)
Domain = x <= 1
c) f o f(x) = (x)^(1/4)
Domain = [0, infinity)
d) g o g(x) = cube root of (1 - cube root of (1-x))
Domain = All real numbers