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12.4.3 Quiz: Combining Functions

Question 4 of 10

Given the functions:

[tex]\[ f(x) = 2x^3 + 5x^2 - 6x - 7 \][/tex]
[tex]\[ g(x) = 2x - 3 \][/tex]

Find [tex]\((f - g)(x)\)[/tex].

A. [tex]\((f - g)(x) = 2x^3 + 5x^2 - 4x + 4\)[/tex]
B. [tex]\((f - g)(x) = 2x^3 + 5x^2 - 4x - 4\)[/tex]
C. [tex]\((f - g)(x) = 2x^3 + 5x^2 - 8x + 4\)[/tex]
D. [tex]\((f - g)(x) = 2x^3 + 5x^2 - 8x - 4\)[/tex]



Answer :

GinnyAnswer
Alright, let's solve the problem step-by-step. We are given two functions:

[tex]\[ f(x) = 2x^3 + 5x^2 - 6x - 7 \][/tex]
[tex]\[ g(x) = 2x - 3 \][/tex]

We need to find [tex]\((f - g)(x)\)[/tex], which means we subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex].

Step-by-Step Solution:

1. Write down the expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]

2. Substitute the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the expression:
[tex]\[ (f - g)(x) = (2x^3 + 5x^2 - 6x - 7) - (2x - 3) \][/tex]

3. Distribute the negative sign across the terms in [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = 2x^3 + 5x^2 - 6x - 7 - 2x + 3 \][/tex]

4. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(2x^3\)[/tex]
- The [tex]\(x^2\)[/tex] term: [tex]\(5x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-6x - 2x = -8x\)[/tex]
- The constant terms: [tex]\(-7 + 3 = -4\)[/tex]

So, combining all the terms, we get:
[tex]\[ (f - g)(x) = 2x^3 + 5x^2 - 8x - 4 \][/tex]

Therefore, the correct answer is:

OD. [tex]\((f - g)(x) = 2x^3 + 5x^2 - 8x - 4\)[/tex]

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