Question

Given the functions, f(x) = 3x - 6 and g(x) = x2 + 5x + 7, perform the indicated operation. When applicable, state the domain restriction.

f(g(x))
A. x2 + 8 x + 1
B. 3 x3 - x2 - 9 x - 42
C. 3 x2 + 15 x + 21
D. 3 x2 + 15 x + 15

Answer 1

f(g(x)), where f(x) = 3x - 6 and g(x) = x^2 + 5x + 7, is calculated by substituting g(x) into f(x). The composition is f(g(x)) = 3x^2 + 15x + 15, which corresponds to answer choice D. There are no domain restrictions for this operation.

Step-by-step explanation:

To calculate the composition of the functions f(g(x)), where f(x) = 3x - 6 and g(x) = x2 + 5x + 7, substitute g(x) into f(x). This means we will replace x in the function f(x) with the entire function g(x).

Start by writing down f(g(x)):

f(g(x)) = 3(g(x)) - 6

Now substitute g(x) into the equation:

f(g(x)) = 3(x2 + 5x + 7) - 6

Then, distribute the 3:

f(g(x)) = 3x2 + 15x + 21 - 6

Simplify by combining like terms:

f(g(x)) = 3x2 + 15x + 15

So, the answer is D. 3x2 + 15x + 15. There are no domain restrictions in this composition because the domain of g(x) does not impose any restrictions on the domain of f(x).