\(\boxed{A. \ f(g(2)) = -7 \ }\)
Further explanation
In this problem we will find out the value of the function composition. There are two ways to do it.
\(\boxed{ \ f(x) = x + 8 \ }\)
\(\boxed{ \ g(x) = x^2 - 6x - 7 \ }\)
\(\boxed{ \ f(g(2)) = ? \ }\)
First way
Step-1: compose (f o g)(x) = f(g(x))
Here g(x) as input into f(x). In other words, first we apply g(x), then apply f(x) to that result:
\(g(x) = x^2 - 6x - 7 \rightarrow f(x) = x+8\)
\(f(g(x)) = (x^2 - 6x - 7) + 8\)
\(f(g(x)) = x^2 - 6x - 7 + 8\)
And we get,
\(\boxed{ \ f(g(x)) = x^2 - 6x + 1 \ }\)
Step-2: calculate the value of f(g(2))
After getting f(g(x)) we proceed by calculating the value f (g(2)).
\(x = 2 \rightarrow f(g(2)) = (2)^2 - 6(2) + 1\)
\(f(g(2)) = 4 - 12 + 1\)
And we obtain the final result:
\(\boxed{ \ f(g(2)) = -7 \ }\)
Second way
Step-1: count g(2) initially
\(x = 2 \rightarrow g(2) = (2)^2 - 6(2) - 7\)
\(g(2) = 4 - 12 - 7\)
And we get,
\(\boxed{ \ g(2) = -15 \ }\)
Step-2: calculate the value of f(g(2))
Here the value of g(2), i.e. -15, as input into f(x).
\(g(2) = -15 \rightarrow f(-15) = -15 + 8\)
And we obtain the final result:
\(\boxed{ \ f(g(2)) = -7 \ }\)
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Keywords: Let f(x) = x + 8 and g(x) = x² - 6x - 7, find f(g(2)), composition function, input, f(g(x)), value, initially,
