To find the domain of the composition of two functions, g(f(x)), we first need to determine the domains of both functions individually.
The function f(x) = x^2 + 1 is a polynomial function, which means it is defined for all real numbers. Therefore, the domain of f(x) is:
- Domain of f(x): (-∞, ∞)
Next, we consider the function g(x) = 2x + 3, which is also a polynomial function. This function is also defined for all real numbers, thus the domain of g(x) is:
- Domain of g(x): (-∞, ∞)
Now we need to find the composite function g(f(x)). For g(f(x)) to be defined, f(x) must be in the domain of g. Since f(x) produces real values for all real x, and g is defined for all real numbers, the output of f(x) will always be a valid input for g(x).
Thus, the domain of g(f(x)) is:
- Domain of g(f(x)): (-∞, ∞)