Answer:
In mathematics, the domain and range of a function are important concepts that describe the set of possible input values (domain) and the corresponding output values (range). When you have a composition of functions, such as (g ∘ f)(x), the domain and range can be affected by both individual functions.
Let's break it down:
Given functions:
f(x)
g(x)
The composition (g ∘ f)(x) means that you first apply the function f to x, and then apply the function g to the result of f(x). Mathematically, this is represented as:
(g ∘ f)(x) = g(f(x))
Now, let's consider the domain and range:
Domain of (g ∘ f)(x):
The domain of (g ∘ f)(x) is determined by the domain of f(x) for which the composition is defined, and then further constrained by the domain of g(x) for which the composed function makes sense.
Range of (g ∘ f)(x):
The range of (g ∘ f)(x) is determined by the range of g(x), as this is the final output of the composition.
It's important to note that the specifics of the domain and range will depend on the actual functions f(x) and g(x) that you're working with. Without knowing these functions, it's not possible to provide precise details about the domain and range of their composition (g ∘ f)(x). If you provide the functions f(x) and g(x), I can help you determine the domain and range for their composition.