Math 102 Quiz 3 Solution: Functions, Domain, Apple Production

This document presents the solutions to Math 102 Quiz 3, focusing on several key mathematical concepts. The first section addresses function operations, including subtraction, composition (f o g and g o f), and evaluating composite functions at a specific point. The second part delves into determini...Read More

Math 102 – 001

Solution1.1:-
f (x) = 3x-1
g (x)= x2-2x
According to the property f-g= (f-g) x=f(x)-g(x)
(f-g) x= 3x-1 - (x2-2x)
(f-g) x= 3x-1 + (x2-2x)
(f-g) x=5x-1- x2 or
(f-g) x= - ( x2 - 5x+1)
1.2 Solution:-
According to the property (fog) x = f(g(x))
=f ((-2x))
=3(x2-2x)-1
=3 x2-6x-1
1.3 Solution:-
According to the property (gof) x = g(f(x))
=g(f(x)
=g(3x-1)
g(x)= x2-2x
= (3x-1)2 -2(3x-1)
= 9x2+1-6x-6x+2
= 9x3+3
1.4 Solution:-
=g (3x-1)
Put x=0 According to the g (f (0))

It becomes g(-1)
Now put -1 in g(x)
G (x) = x2-2x
g (-1) = (-1)2 -2(-1)
= 1+2
= 3
2.1Solution:-
f (x) = √ (6-x)
Domain =√ (6-x): x<=6
Intervals: (-infinity, 6)
6-x>=0
-x>=-6
Therefore Non- negative or positive value exist x<=6
Case 1:-(-infinity, -6) =√ (6-x) put x=-5
=6-x = 6(-5) = 6+5 =11
Case 2:-(-6, 6) =6-x put x=5
=6-5
=1
Case 3 :-(6, infinity) ==6-x put x=7
=-1
Domain x<=6

2.2 Solution:-
g(x) = x^4-3x^3+5x^2-2
x^4-3x^3+5x^2-2 lies in –infinity to infinity
Interval: (–infinity, infinity)
The functions have not under defined points and no any domain constraints. Because it’s real roots not
exist.
Therefore domain: -infinity<x< infinity
3 Solution:-
Each trees produces Apples = 900 - 9n
An =n (900-9n)
An=900n-9n2
An=-9( 100n+n2)
0r
An=-9(n2 +100n)
On Solving
An=-9(n2 +100n+ (-50)2) + 9+ 9( -50)2
An =-9(n-50)2
The Maximum Number of apple produced each year per year = 22500 apples
4 Solution ( Bonus Question):-
f (x) = 3x-1
put the value in ( f( x + h) – f(x)/h)
((2 x -3 +h)-(2x-3) /h)
(2x-3 +h -2h +3)/h
=h/h
=1