There are three parts to this assignment.
Part A is to be completed online before 7:00 a.m. on Friday,
October 1.
Part B and Part C , which require full solutions, are to be handed in at the beginning of class on the same date.
Part A [10 marks]
This part of the assignment can be found online, labelled A2 , at webwork.elearning.ubc.ca
— sign in using the MATH110 001 2010W button.
Part B [5 marks]
This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions.
From the Calculus: Early Transcendentals text, complete questions 62 and 63 from section 1.2.
Part C [15 marks]
This part of the assignment consists of more challenging questions. You are expected to provide full solutions with complete arguments and justifications.
1. A function f is even if f ( x ) = f ( − x ) for all x , and odd if f ( x ) = − f ( − x ) for all x . For example, f ( x ) = x 2 is even, and f ( x ) = x 3 is odd.
Prove that the composition of two even functions is even, and that the composition of two odd functions is odd.
2. A function f is injective if every element in the range is mapped to by exactly one element in the domain; that is, if f ( x
1
) = f ( x
2
) only when x
1
= x
2
. For example, f ( x ) = x is injective; f ( x ) = x 2 is not. A function f is invertible if there exists a function g that “reverses” f ; that is, that satisfies
( f ◦ g )( x ) = ( g ◦ f )( x ) = x for all x .
For example, f ( x ) = x 3 is invertible; its inverse is g ( x ) = x 1 / 3 .
Prove that an invertible function must also be injective.
3. A number a is a fixed point of a function f if f ( a ) = a . For example, 1 is a fixed point of f ( x ) = x 2 .
Prove that any cubic function f ( x ) = Ax
3
+ Bx
2
+ Cx + D must have a fixed point.