Let's tackle this question step-by-step, ensuring we understand each part clearly.
Part A: What is (f + g)(x)? Show all necessary steps.
To find [tex]\((f + g)(x)\)[/tex], we need to add the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together:
Given:
[tex]\[ f(x) = x + 5 \][/tex]
[tex]\[ g(x) = 4x + 7 \][/tex]
[tex]\((f + g)(x)\)[/tex] represents the sum of these two functions:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
Substitute the given functions into this equation:
[tex]\[
(f + g)(x) = (x + 5) + (4x + 7)
\][/tex]
Combine like terms:
[tex]\[
(f + g)(x) = x + 4x + 5 + 7
\][/tex]
[tex]\[
(f + g)(x) = 5x + 12
\][/tex]
So, [tex]\((f + g)(x) = 5x + 12\)[/tex].
Part B: Evaluate (f + g)(4). Show your work.
To evaluate [tex]\((f + g)(4)\)[/tex], we substitute [tex]\(x = 4\)[/tex] into the function [tex]\((f + g)(x)\)[/tex]:
[tex]\[
(f + g)(4) = 5(4) + 12
\][/tex]
Calculate the product and sum:
[tex]\[
(f + g)(4) = 20 + 12
\][/tex]
[tex]\[
(f + g)(4) = 32
\][/tex]
So, [tex]\((f + g)(4) = 32\)[/tex].
Part C: Explain what your answer represents in terms of the scenario.
In the context of the scenario provided, [tex]\(f(x)\)[/tex] represents the number of animals adopted at one shelter after [tex]\(x\)[/tex] months, while [tex]\(g(x)\)[/tex] represents the number of animals adopted at another shelter after [tex]\(x\)[/tex] months.
When we evaluate [tex]\((f + g)(4)\)[/tex], we are finding the total number of animals adopted by both shelters combined after 4 months. Therefore, the result [tex]\(32\)[/tex] means that a total of 32 animals have been adopted by both shelters together after a period of 4 months.
