Let
f(x) = square root (x + 2)
and
g(x) = x^2 − 2.
Determine the domain of the composite function. (Enter your answer using interval notation.)
f compose f
______?
& Find the composite function.
(f compose f)(x) = _____?
To determine the domain of the composite function f composed with f, we need to consider the input values that are valid for both functions.
Let's first find the composite function:
(f compose f)(x) = f(f(x))
To compose the two functions, we substitute the function g into f as follows:
(f compose f)(x) = f(g(x))
Substituting g(x) = x^2 - 2 into f(x) = √(x + 2), we get:
(f compose f)(x) = f(g(x)) = f(x^2 - 2) = √((x^2 - 2) + 2) = √(x^2) = |x|
So, the composite function is (f compose f)(x) = |x|.
Now, let's determine the domain of the composite function.
The domain of the composite function f composed with f is the set of values of x that are valid for both f(x) and f(f(x)), or in this case, f(x) = √(x + 2) and (f compose f)(x) = |x|.
Since there are no restrictions on taking square roots or absolute values, the domain of the composite function is the set of all real numbers, which can be represented in interval notation as (-∞, +∞).