Question

If f(x) = 2x + 7 and g(x) = x², find the following.

Answer 1

The student's question pertains to finding the result of combining two given mathematical functions. Without a specific operation provided in the question, one common combination method, function composition, was demonstrated. The composition (f ∘ g)(x) was found to be f(g(x)) = 2x² + 7.

Step-by-step explanation:

The student is asking to find the functions obtained by combining the given functions f(x) = 2x + 7 and g(x) = x². However, the actual operations to be performed (like function composition, addition, etc.) with these functions are not specified in the question provided. Therefore, I'll explain how to do function composition as an example, which is a common operation involving two functions.

Function Composition:

To find the composition of the two functions, denoted as (f ∘ g)(x) or f(g(x)), we substitute g(x) into f(x). Here's how it's done:

Start with the function f(x) = 2x + 7.

Replace the x in f(x) with g(x) = x², getting f(g(x)) = 2(x²) + 7.

Simplify this expression to f(g(x)) = 2x² + 7.

If the student meant a different operation like addition, subtraction, or another function operation, they will need to specify this, and the process would differ.

Answer 2

The function f(g(x)) equals 2x² + 7, and the function g(f(x)) equals (2x + 7)². These functions are compositions of f(x) = 2x + 7 and g(x) = x².

Step-by-step explanation:

To find the values of the functions f(x) = 2x + 7 and g(x) = x² when composed, we perform the following steps:

1. For f(g(x)): Substitute x² for x in the function f(x):
   
2. For g(f(x)): Substitute (2x + 7) for x in the function g(x):
   f(g(x)) results in the expression

2x² + 7

, and g(f(x)) results in the expression

(2x + 7)²

, which can further be expanded and simplified if necessary.