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 determining the domain of given functions, including a square root function and a polynomial function. Finally, the assignment tackles a word problem related to an apple orchard, where students are tasked with finding the optimal number of trees to plant per acre to maximize apple yield. The solution involves setting up a quadratic equation and finding its maximum value, providing a practical application of mathematical principles.

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