November 3rd, 2014, 02:19 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! 
November 3rd, 2014, 03:56 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,118 Thanks: 2369 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 $f^{1}(f(x)) = x$. 

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