To find the value of [tex]\( f(-5) + g(-5) \)[/tex] given the functions [tex]\( f(x) = 3x + 5^2 \)[/tex] and [tex]\( g(x) = -x^3 - 3x^2 - x \)[/tex], we'll follow these steps:
1. Calculate [tex]\( f(-5) \)[/tex]:
- First, plug in [tex]\( x = -5 \)[/tex] into the function [tex]\( f(x) \)[/tex].
- The function [tex]\( f(x) \)[/tex] is defined as [tex]\( f(x) = 3x + 5^2 \)[/tex].
- Simplify the expression inside the function:
[tex]\[
f(-5) = 3(-5) + 5^2
\][/tex]
- Calculate [tex]\( 3(-5) \)[/tex]:
[tex]\[
3(-5) = -15
\][/tex]
- Calculate [tex]\( 5^2 \)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
- Combine these results:
[tex]\[
f(-5) = -15 + 25 = 10
\][/tex]
- Therefore, [tex]\( f(-5) = 10 \)[/tex].
2. Calculate [tex]\( g(-5) \)[/tex]:
- Next, plug in [tex]\( x = -5 \)[/tex] into the function [tex]\( g(x) \)[/tex].
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = -x^3 - 3x^2 - x \)[/tex].
- Simplify the expression inside the function:
[tex]\[
g(-5) = -(-5)^3 - 3(-5)^2 - (-5)
\][/tex]
- Calculate [tex]\((-5)^3 \)[/tex]:
[tex]\[
(-5)^3 = -125
\][/tex]
- Calculate [tex]\(-(-5)^3 \)[/tex]:
[tex]\[
-(-125) = 125
\][/tex]
- Calculate [tex]\((-5)^2 \)[/tex]:
[tex]\[
(-5)^2 = 25
\][/tex]
- Calculate [tex]\( -3(-5)^2 \)[/tex]:
[tex]\[
-3(25) = -75
\][/tex]
- Simplify the term [tex]\( -(-5) \)[/tex]:
[tex]\[
-(-5) = 5
\][/tex]
- Combine these results:
[tex]\[
g(-5) = 125 - 75 + 5 = 55
\][/tex]
- Therefore, [tex]\( g(-5) = 55 \)[/tex].
3. Find [tex]\( f(-5) + g(-5) \)[/tex]:
- Now, add the results from [tex]\( f(-5) \)[/tex] and [tex]\( g(-5) \)[/tex]:
[tex]\[
f(-5) + g(-5) = 10 + 55 = 65
\][/tex]
Therefore, the value of [tex]\( f(-5) + g(-5) \)[/tex] is [tex]\( 65 \)[/tex].
