Let's evaluate the function [tex]\( f(x) = x^2 - 2x - 5 \)[/tex] at the given values and simplify the expressions step-by-step.
### Part (a): Evaluate [tex]\( f(-7) \)[/tex]
To find [tex]\( f(-7) \)[/tex], we substitute [tex]\( x = -7 \)[/tex] into the function:
[tex]\[
f(-7) = (-7)^2 - 2(-7) - 5
\][/tex]
Simplify each term:
- [tex]\((-7)^2 = 49\)[/tex]
- [tex]\(-2(-7) = 14\)[/tex]
- [tex]\(-5\)[/tex] remains [tex]\(-5\)[/tex]
Combine these results:
[tex]\[
f(-7) = 49 + 14 - 5 = 63 - 5 = 58
\][/tex]
So,
[tex]\[
f(-7) = 58
\][/tex]
### Part (b): Evaluate [tex]\( f(x+7) \)[/tex]
To find [tex]\( f(x+7) \)[/tex], we substitute [tex]\( x = x+7 \)[/tex] into the function:
[tex]\[
f(x+7) = (x+7)^2 - 2(x+7) - 5
\][/tex]
First, expand [tex]\( (x+7)^2 \)[/tex]:
[tex]\[
(x+7)^2 = x^2 + 14x + 49
\][/tex]
Next, distribute [tex]\(-2\)[/tex] to [tex]\((x+7)\)[/tex]:
[tex]\[
-2(x+7) = -2x - 14
\][/tex]
Combine all these terms:
[tex]\[
f(x+7) = x^2 + 14x + 49 - 2x - 14 - 5
\][/tex]
Simplify by combining like terms:
[tex]\[
f(x+7) = x^2 + (14x - 2x) + (49 - 14 - 5) = x^2 + 12x + 30
\][/tex]
Thus,
[tex]\[
f(x+7) = x^2 + 12x + 30
\][/tex]
### Part (c): Evaluate [tex]\( f(-x) \)[/tex]
To find [tex]\( f(-x) \)[/tex], we substitute [tex]\( x = -x \)[/tex] into the function:
[tex]\[
f(-x) = (-x)^2 - 2(-x) - 5
\][/tex]
Simplify each term:
- [tex]\((-x)^2 = x^2\)[/tex]
- [tex]\(-2(-x) = 2x\)[/tex]
- [tex]\(-5\)[/tex] remains [tex]\(-5\)[/tex]
Combine these results:
[tex]\[
f(-x) = x^2 + 2x - 5
\][/tex]
Thus,
[tex]\[
f(-x) = x^2 + 2x - 5
\][/tex]
### Summary:
a. [tex]\( f(-7) = 58 \)[/tex]
b. [tex]\( f(x+7) = x^2 + 12x + 30 \)[/tex]
c. [tex]\( f(-x) = x^2 + 2x - 5 \)[/tex]
