To determine the domain of the composite function g(f(x)), we first need to understand the individual domains of the functions f(x) and g(x) and how they interact.
The function f(x) = x^2 + 1 is a polynomial function. Polynomial functions are defined for all real numbers, so the domain of f(x) is:
- Domain of f(x): All real numbers, denoted as (-∞, ∞).
Next, let’s analyze the function g(x) = 2x + 3. This is also a polynomial function, and like all polynomial functions, it is defined for all real numbers as well. Therefore, the domain of g(x) is:
- Domain of g(x): All real numbers, denoted as (-∞, ∞).
Now, we want to find the domain of the composite function g(f(x)). For the composite function g(f(x)) to be defined, the output of f(x) must lie within the domain of g(x).
Since f(x) produces all real numbers (as its output, x^2 + 1, is always ≥ 1 for all real x), the output of f(x) definitely falls within the domain of g(x). Thus, there are no restrictions placed by g(x) on the inputs of f(x).
As a result, we can conclude that the domain of g(f(x)) is:
- Domain of g(f(x)): All real numbers, denoted as (-∞, ∞).
In summary, the composite function g(f(x)) is defined for all real numbers, making its domain the same as that of both f(x) and g(x).