Question

Problem Set 1: IA 4 b and c:
I'm confused on even and odd equations, I guess. Can anyone further explain what the difference is? (... They're dividing by two to get f(x) instead of 2f(x) in b, I get that, mostly, but in c, how are they finding out if they're even or odd? Is it just intuitive b/c x cubed of an odd will = an odd number?)

Answer 1

it's very simple
just replace every (x) with (-x)
if it's the same equation (f(-x)=f(x)) then it's even
if it equals negative the original equation (f(-x)=-f(x)) then it's odd

Answer 2

x cubed of an odd will = an odd number?
that motivates the names: contrast the graph of x^2 and x^3

notice for y= x^2
for x= 1 or -1 we get y=1
or , in general, for \( x= \pm x_0\) , y = \( x_0^2\)
we can write this as
f(x) = f(-x)
and this is the definition of an even function.

similarly,
f(x) = - f(-x)
is the definition of an odd function.

Answer 3

Functions are classified into even/odd for a better understanding of their symmetry. If a function of x is even, then its graph on the Cartesian co-ordinates would be symmetric about the Y axis; i.e. the Y axis splits the curve into two equal and congruent halves.
For example if f(2) and f(-2) exist then this would imply, f(2) = f(-2).
For any general value of x, check if f(x) = f(-x) OR f(x) - f(-x) = 0

If a function of x is odd then its graph would be symmetric about the origin; i.e. there will be equal and congruent halves of the curve in opposite quadrants.
For example if f(1) and f(-1) exist then according to the symmetry condition it can be said that f(-1) = - f(1).
For any general value of x, check if f(-x) = -f(x) OR f(x)- f(-x) = 0