If f(x) = x² + 25 and g(x) = x + 5, what is the domain of f(g(x)) and find f(g(1))?

To determine the domain of the composite function f(g(x)), we first need to look at the domains of both functions f(x) and g(x).

The function f(x) = x² + 25 is a polynomial function and is defined for all real numbers, meaning its domain is (−∞, ∞).

Next, let’s consider the function g(x) = x + 5. This is also a polynomial function and, like f(x), it is defined for all real numbers as well. Therefore, g(x) also has a domain of (−∞, ∞).

Since g(x) is defined for all real numbers and f(x) is defined for all real numbers, the composite function f(g(x)) will also be defined for all real numbers. Consequently, the domain of f(g(x)) is (−∞, ∞).

Now, let’s find f(g(1)).

First, we calculate g(1):

g(1) = 1 + 5 = 6

Now we will substitute this result into the function f(x):

f(g(1)) = f(6) = 6² + 25 = 36 + 25 = 61

Thus, f(g(1)) = 61.