To find the results of combining the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] for a specific value of [tex]\(x\)[/tex], we will evaluate the functions for [tex]\(x = 4\)[/tex] and then compute the combined functions [tex]\((f+g)(x)\)[/tex], [tex]\((f-g)(x)\)[/tex], and [tex]\((f \times g)(x)\)[/tex].
### Step-by-Step Solution:
1. Define the Functions:
[tex]\[
f(x) = x^2 + 6x + 9
\][/tex]
[tex]\[
g(x) = x^2 - 9
\][/tex]
2. Evaluate [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] at [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = 4^2 + 6(4) + 9 = 16 + 24 + 9 = 49
\][/tex]
[tex]\[
g(4) = 4^2 - 9 = 16 - 9 = 7
\][/tex]
3. Combine the Functions:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
[tex]\[
(f \times g)(x) = f(x) \times g(x)
\][/tex]
4. Evaluate the Combinations at [tex]\(x = 4\)[/tex]:
[tex]\[
(f + g)(4) = f(4) + g(4) = 49 + 7 = 56
\][/tex]
[tex]\[
(f - g)(4) = f(4) - g(4) = 49 - 7 = 42
\][/tex]
[tex]\[
(f \times g)(4) = f(4) \times g(4) = 49 \times 7 = 343
\][/tex]
### Matching Each Combination to the Value:
1. [tex]\((f+g)(4) = 56\)[/tex]
2. [tex]\((f-g)(4) = 42\)[/tex]
3. [tex]\((f \times g)(4) = 343\)[/tex]
Thus, the results for the combined functions are:
[tex]\(
(f+g)(4) = 56 \\
(f-g)(4) = 42 \\
(f \times g)(4) = 343 \\
\)[/tex]
Each combination is matched to its corresponding value as:
- [tex]\((f + g)(4) \rightarrow 56\)[/tex]
- [tex]\((f - g)(4) \rightarrow 42\)[/tex]
- [tex]\((f \times g)(4) \rightarrow 343\)[/tex]
So, ensuring that these combinations are correctly matched with the values given:
1. [tex]\( (f+g)(x) \rightarrow 6(x+3) \rightarrow 56 \)[/tex]
2. [tex]\( (f-g)(x) \rightarrow 2x(x+3) \rightarrow 42\)[/tex]
3. [tex]\( (f\times g)(x) \rightarrow x^4+6x^3-54 x-81 \rightarrow 343\)[/tex]
