Given the functions [tex]f(x) = -7x^5 + 9x^7 + 1 + 3x^6[/tex] and [tex]g(x) = 2 - 5x^6 + 7x^7 - 2x^4[/tex], find [tex](f + g)(x)[/tex] and [tex](f + g)(1)[/tex].

To find [tex](f + g)(x)[/tex], combine the functions [tex]f(x)[/tex] and [tex]g(x)[/tex]:

[tex](f + g)(x) = (-7x^5 + 9x^7 + 1 + 3x^6) + (2 - 5x^6 + 7x^7 - 2x^4)[/tex]

Now simplify the expression.

To solve the problem, we need to find the sum of two polynomial functions [tex]$ f(x) $[/tex] and [tex]$ g(x) $[/tex], which are given as:

1. [tex]$ f(x) = -7x^5 + 9x^7 + 1 + 3x^6 $[/tex]
2. [tex]$ g(x) = 2 - 5x^6 + 7x^7 - 2x^4 $[/tex]

### Step-by-Step Solution:

1. Find [tex]$(f + g)(x)$[/tex]:

To find the sum [tex]$(f + g)(x)$[/tex], we combine like terms from both [tex]$ f(x) $[/tex] and [tex]$ g(x) $[/tex]:

[tex]\[
f(x) = -7x^5 + 9x^7 + 1 + 3x^6
\][/tex]

[tex]\[
g(x) = 2 - 5x^6 + 7x^7 - 2x^4
\][/tex]

We add these polynomials by combining the coefficients of the same powers of [tex]$ x $[/tex]:

- [tex]$ x^7 $[/tex] term: [tex]$ 9x^7 + 7x^7 = 16x^7 $[/tex]
- [tex]$ x^6 $[/tex] term: [tex]$ 3x^6 - 5x^6 = -2x^6 $[/tex]
- [tex]$ x^5 $[/tex] term: Only in [tex]$ f(x) $[/tex], so it's [tex]$-7x^5$[/tex]
- [tex]$ x^4 $[/tex] term: Only in [tex]$ g(x) $[/tex], so it's [tex]$-2x^4$[/tex]
- Constant term: [tex]$ 1 + 2 = 3 $[/tex]

Combining all, we get:

[tex]\[
(f + g)(x) = 16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]

2. Evaluate [tex]$(f + g)(1)$[/tex]:

Now, substitute [tex]$ x = 1 $[/tex] into [tex]$(f + g)(x)$[/tex]:

[tex]\[
(f + g)(1) = 16(1)^7 - 2(1)^6 - 7(1)^5 - 2(1)^4 + 3
\][/tex]

Simplify this expression:

- [tex]$ 16(1) - 2(1) - 7(1) - 2(1) + 3 = 16 - 2 - 7 - 2 + 3 $[/tex]
- Combine the terms: [tex]$ 16 - 2 - 7 - 2 + 3 = 8 $[/tex]

Therefore, [tex]$(f + g)(1) = 8$[/tex].

So the final polynomial [tex]$(f + g)(x)$[/tex] is:

[tex]\[
16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]

And [tex]$(f + g)(1)$[/tex] evaluates to 8.