My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 3rd, 2014, 01:19 PM   #1
Senior Member
 
Joined: Oct 2014
From: Complex Field

Posts: 119
Thanks: 4

This question is too hard for me

Hello,
I got a question in my homework with 6 sections to prove or disprove
Can you help me and also with the mathematical symbols? I mean how to write this exactly? It's a bit hard question so even if you prove/disprove just 1 of out 6 it would be helpful.

The question goes:

Prove or disprove with a counter example:

(1) f, g : R → R are even functions => f ∘ g is even.
(2) f : R → R is even , g : R → R is odd => f ∘ g is even.
(3) g : R → R is even , f : R → R can be any function => f ∘ g is even
(4) f, g : R → R are monotonically increasing functions => f ∘ g is a monotonically increasing function.
(5) f, g : R → R are monotonically decreasing functions => f ∘ g is a monotonically decreasing function.
(6) f : R → R is an inversible and monotonically increasing function => f^-1 (the inverse function) is monotonically increasing function.


*Number 6 was particularly difficult.

Thank you all the forum genius guys and girls!
noobinmath is offline  
 
November 3rd, 2014, 02:56 PM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,327
Thanks: 2451

Math Focus: Mainly analysis and algebra
An even function $f(x)$ has: $f(-x) = f(x)$
An odd function $f(x)$ has: $f(-x) = -f(x)$
A monotonically increasing function $y = f(x)$ has: $f(x + \Delta x) = y + \Delta y$ where $\Delta x \gt 0$ and $\Delta y \ge 0$
A monotonically increasing function $y = f(x)$ has: $f(x + \Delta x) = y - \Delta y$ where $\Delta x \gt 0$ and $\Delta y \ge 0$

Use those equalities in the definitions given. e.g. the first question:$$f \circ g(-x) = f(g(-x)) = f(g(x)) = f \circ g(x)$$

For the last one, you may want to use $$g(f(x)) = x \; \forall x \implies g^\prime(x) = \frac{1}{f^\prime(x)}$$if you have learned it. If you haven't learned it yet, you'll need to use something like $f^{-1}(f(x)) = x$.
v8archie is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
hard, question



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
very hard question riotsandravess Advanced Statistics 4 November 3rd, 2010 06:22 PM
very hard question riotsandravess Advanced Statistics 0 October 28th, 2010 01:21 PM
Very hard question mathhelp123 Calculus 21 October 6th, 2009 05:21 AM
Hard Question for Me Biggzi Algebra 2 February 4th, 2009 09:23 AM
really Hard question helpmewithmath Algebra 3 November 14th, 2007 10:12 AM





Copyright © 2018 My Math Forum. All rights reserved.