Evaluate the function [tex]f(x) = x^2 + 9x + 1[/tex] at the given values of the independent variable and simplify.

a. [tex]f(8)[/tex]

b. [tex]f(x+5)[/tex]

c. [tex]f(-x)[/tex]

- [tex]f(8) = 137[/tex] (Simplify your answer.)

- [tex]f(x+5) = x^2 + 19x + 71[/tex] (Simplify your answer.)

- [tex]f(-x) = \square[/tex] (Simplify your answer.)

Sure! Let's evaluate the function [tex]$ f(x) = x^2 + 9x + 1 $[/tex] for each of the given values and simplify.

### a. [tex]$ f(8) $[/tex]
To evaluate [tex]$ f(8) $[/tex], substitute [tex]$ x = 8 $[/tex] into the function:

[tex]\[
f(8) = 8^2 + 9 \cdot 8 + 1
\][/tex]

[tex]\[
= 64 + 72 + 1
\][/tex]

[tex]\[
= 137
\][/tex]

So, [tex]$ f(8) = 137 $[/tex].

### b. [tex]$ f(x+5) $[/tex]
To evaluate [tex]$ f(x+5) $[/tex], substitute [tex]$ x+5 $[/tex] into the function:

[tex]\[
f(x+5) = (x+5)^2 + 9(x+5) + 1
\][/tex]

First, expand [tex]$ (x+5)^2 $[/tex]:

[tex]\[
(x+5)^2 = x^2 + 10x + 25
\][/tex]

Now substitute back:

[tex]\[
f(x+5) = x^2 + 10x + 25 + 9(x+5) + 1
\][/tex]

Expand [tex]$ 9(x+5) $[/tex]:

[tex]\[
9(x+5) = 9x + 45
\][/tex]

Combine the terms:

[tex]\[
f(x+5) = x^2 + 10x + 25 + 9x + 45 + 1
\][/tex]

Combine like terms:

[tex]\[
= x^2 + (10x + 9x) + (25 + 45 + 1)
\][/tex]

[tex]\[
= x^2 + 19x + 71
\][/tex]

So, [tex]$ f(x+5) = x^2 + 19x + 71 $[/tex].

### c. [tex]$ f(-x) $[/tex]
To evaluate [tex]$ f(-x) $[/tex], substitute [tex]$-x$[/tex] into the function:

[tex]\[
f(-x) = (-x)^2 + 9(-x) + 1
\][/tex]

Calculate [tex]$(-x)^2$[/tex]:

[tex]\[
(-x)^2 = x^2
\][/tex]

Now substitute back:

[tex]\[
f(-x) = x^2 - 9x + 1
\][/tex]

So, [tex]$ f(-x) = x^2 - 9x + 1 $[/tex].

These are the simplified results for each part of the question:
- [tex]$ f(8) = 137 $[/tex]
- [tex]$ f(x+5) = x^2 + 19x + 71 $[/tex]
- [tex]$ f(-x) = x^2 - 9x + 1 $[/tex]