Let
f(x) = x + 2
and
g(x) = x2 − 2.
Determine the domain of the composite function.
(Enter your answer using interval notation.)
f compose f
Whether you want f(g),g(f),f(f) or g(g), the composite will be a polynomial. The domain of all polynomials is (-∞,∞)
To determine the domain of the composite function, we need to consider the domains of the individual functions involved.
The function f(x) = x + 2 is a linear function, which means it is defined for all real numbers. Therefore, the domain of f(x) is the set of all real numbers.
The function g(x) = x^2 − 2 is a quadratic function, and since it is a polynomial function, it is defined for all real numbers. So, the domain of g(x) is also the set of all real numbers.
To find the domain of the composite function f(g(x)), we need to consider the restrictions that arise from the composition of the two functions. In this case, we need to consider any values of x that result in undefined outputs when evaluating f(g(x)).
When we plug g(x) into f(x), we get f(g(x)) = g(x) + 2 = (x^2 - 2) + 2 = x^2.
It means that the composite function f(g(x)) reduces to a simple quadratic function x^2, which is defined for all real numbers.
Therefore, the domain of the composite function f(g(x)) is also the set of all real numbers.