Let's solve the given operations step by step for the functions f(x)=3x+2x2−10 and g(x)=5x−x2+15.
A. (f+g)(x)
This is simply adding the two functions:
(f+g)(x)=f(x)+g(x)
f(x)=3x+2x2−10
g(x)=5x−x2+15
Adding them together:
(f+g)(x)=(3x+2x2−10)+(5x−x2+15)
=3x+5x+2x2−x2−10+15
=8x+x2+5
So, (f+g)(x)=x2+8x+5.
B. (f+g)(5)
Now, substitute x=5 into (f+g)(x):
(f+g)(5)=52+8(5)+5
=25+40+5
=70
So, (f+g)(5)=70.
C. (f−g)(x)
Subtract the functions:
(f−g)(x)=f(x)−g(x)
=(3x+2x2−10)−(5x−x2+15)
=3x−5x+2x2−(−x2)−10−15
=−2x+3x2−25
So, (f−g)(x)=3x2−2x−25.
D. (f−g)(−6)
Now, substitute x=−6 into (f−g)(x):
(f−g)(−6)=3(−6)2−2(−6)−25
=3(36)+12−25
=108+12−25
=95
So, (f−g)(−6)=95.
E. (f⋅g)(x)
Multiply the functions:
(f⋅g)(x)=f(x)⋅g(x)
=(3x+2x2−10)⋅(5x−x2+15)
We can expand this product using distribution:
=(3x)(5x)+(3x)(−x2)+(3x)(15)+(2x2)(5x)+(2x2)(−x2)+(2x2)(15)+(−10)(5x)+(−10)(−x2)+(−10)(15)
=15x2−3x3+45x+10x3−2x4+30x2−50x+10x2−150
=−2x4+7x3+55x2−5x−150
So, (f⋅g)(x)=−2x4+7x3+55x2−5x−150.
F. (f⋅g)(2)
Substitute x=2 into (f⋅g)(x):
(f⋅g)(2)=−2(2)4+7(2)3+55(2)2−5(2)−150
=−2(16)+7(8)+55(4)−10−150
=−32+56+220−10−150
=84
So, (f⋅g)(2)=84.
G. (f/g)(x)
This is dividing the functions:
(f/g)(x)=g(x)f(x)=5x−x2+153x+2x2−10
H. (f/g)(3)
Substitute x=3 into the quotient:
f(3)=3(3)+2(3)2−10=9+18−10=17
g(3)=5(3)−(3)2+15=15−9+15=21
(f/g)(3)=2117
So, (f/g)(3)=2117.
Would you like further details on any of these steps? Here are 5 related questions to expand your understanding:
- How do you find the composition of two functions?
- What is the inverse of a function, and how is it computed?
- Can you graph (f+g)(x) and (f−g)(x) to compare their shapes?
- What happens when g(x)=0 in (f/g)(x)?
- How do you differentiate and integrate the product of two functions?
Tip: Always remember to check for values where g(x)=0 when performing division of functions, as division by zero is undefined.