Sure! Let's evaluate the function [tex]\( f(x) = x^2 + 3x + 7 \)[/tex] at the given values and simplify step-by-step.
### a. Evaluating [tex]\( f(-6) \)[/tex]
To find [tex]\( f(-6) \)[/tex], substitute [tex]\( x = -6 \)[/tex] into the function:
[tex]\[ f(-6) = (-6)^2 + 3(-6) + 7 \][/tex]
Now calculate each term:
[tex]\[ (-6)^2 = 36 \][/tex]
[tex]\[ 3(-6) = -18 \][/tex]
[tex]\[ 7 = 7 \][/tex]
Adding these together:
[tex]\[ f(-6) = 36 - 18 + 7 = 25 \][/tex]
So, [tex]\( f(-6) = 25 \)[/tex].
### b. Evaluating [tex]\( f(x + 8) \)[/tex]
To find [tex]\( f(x + 8) \)[/tex], substitute [tex]\( x = x + 8 \)[/tex] into the function:
[tex]\[ f(x + 8) = (x + 8)^2 + 3(x + 8) + 7 \][/tex]
Now expand and simplify the terms:
[tex]\[
(x + 8)^2 = x^2 + 16x + 64
\][/tex]
[tex]\[
3(x + 8) = 3x + 24
\][/tex]
[tex]\[
7 = 7
\][/tex]
Combine these:
[tex]\[
f(x + 8) = x^2 + 16x + 64 + 3x + 24 + 7
\][/tex]
Simplify by combining like terms:
[tex]\[
f(x + 8) = x^2 + 19x + 95
\][/tex]
So, [tex]\( f(x + 8) = x^2 + 19x + 95 \)[/tex].
### c. Evaluating [tex]\( f(-x) \)[/tex]
To find [tex]\( f(-x) \)[/tex], substitute [tex]\( x = -x \)[/tex] into the function:
[tex]\[ f(-x) = (-x)^2 + 3(-x) + 7 \][/tex]
Now calculate each term:
[tex]\[
(-x)^2 = x^2
\][/tex]
[tex]\[
3(-x) = -3x
\][/tex]
[tex]\[
7 = 7
\][/tex]
So, combine these:
[tex]\[
f(-x) = x^2 - 3x + 7
\][/tex]
So, [tex]\( f(-x) = x^2 - 3x + 7 \)[/tex].
In summary:
- [tex]\( f(-6) = 25 \)[/tex]
- [tex]\( f(x + 8) = x^2 + 19x + 95 \)[/tex]
- [tex]\( f(-x) = x^2 - 3x + 7 \)[/tex]
