To find the domain of the composite function fg(x), we first need to determine the domains of the individual functions f(x) and g(x).
Step 1: Determine the domain of g(x)
Given g(x) = x + 5, this is a linear function. The domain of g(x) is all real numbers, represented as:
Domain of g(x): (-∞, ∞)
Step 2: Determine the domain of f(x)
We have f(x) = x² + 25. This is a quadratic function, which is also defined for all real numbers. Therefore, its domain is:
Domain of f(x): (-∞, ∞)
Step 3: Find the domain of fg(x)
To find the domain of the composite function fg(x), we need to consider the output of g(x) as it is used as the input for f(x). Since both g(x) and f(x) have the domain of all real numbers, the composite function fg(x) is also defined for all real numbers:
Domain of fg(x): (-∞, ∞)
Step 4: Calculate fg(1)
Now we compute fg(1). This involves first finding g(1) and then using that result in f(x):
1. Calculate g(1):
g(1) = 1 + 5 = 6
2. Now use this result in f:
f(g(1)) = f(6) = 6² + 25 = 36 + 25 = 61
Conclusion
The domain of the composite function fg(x) is (-∞, ∞), and the value of fg(1) is 61.