Solve the following problem:

The function [tex]\( f(x) = x + 5 \)[/tex] represents the number of animals adopted at one shelter, where [tex]\( x \)[/tex] is the number of months.

The function [tex]\( g(x) = 4x + 7 \)[/tex] represents the number of animals adopted at another shelter, where [tex]\( x \)[/tex] is the number of months.

Part A: What is [tex]\( (f + g)(x) \)[/tex]? Show all necessary steps. (2 points)

Part B: Evaluate [tex]\( (f + g)(4) \)[/tex]. Show your work. (2 points)

Part C: Explain what your answer represents in terms of the scenario. (2 points)

Question

29-07-2024
Mathematics

Answered

Solve the following problem:

The function [tex]\( f(x) = x + 5 \)[/tex] represents the number of animals adopted at one shelter, where [tex]\( x \)[/tex] is the number of months.

The function [tex]\( g(x) = 4x + 7 \)[/tex] represents the number of animals adopted at another shelter, where [tex]\( x \)[/tex] is the number of months.

Part A: What is [tex]\( (f + g)(x) \)[/tex]? Show all necessary steps. (2 points)

Part B: Evaluate [tex]\( (f + g)(4) \)[/tex]. Show your work. (2 points)

Part C: Explain what your answer represents in terms of the scenario. (2 points)

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-29T13:12:28-04:00

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.