Can someone explain what this means?
f(-x) = - f(x)
I know that this means if you put something negative then it will be positive. But I dont understand how the equation means this.
And when you flip the equation, it is f(x) = -f(-x) and I dont really understand this either.
Please help me 
Your question doesnāt make any sense.
Even function implies that the graph is unchanged after a reflection over the y axis.
If you consider a point then this is the mapping you expect:
(x,y) \mapsto (-x,y)
This essentially clarifies the f(x) = f(-x) bit.
Same reasoning holds for odd functions, but the self-symmetry is with respect to the origin.
Nope, thatās not the correct interpretation.
Look at whatās happening to the input x on one side, and the output f(x) on the other side.
On the LHS, the input x is being negated to -x
On the RHS, the output f(x) is being negated to -f(x).
So what it means is āif you flip the sign of the input, you get the same output but negatedā. I think you should review this topic entirely again.
What about this one? Unlike the other ones, for the output, there is a negative inside with the x as well so Iām unsure what this means.
Consider f(x) = x^3 then compute f(-1) and f(1).
f(-1) = -1 whereas f(1) = 1
So f(-1) = -f(1)
The implication here is that this self symmetry would hold for all x in the domain of a real odd function of concern
Basically, a real function f is odd if for all x in its domain it satisfies:
f(-x) = -f(x) \implies -f(-x) = f(x)
@hiiie28 does the explanation from @cylverixxx make sense?
I think so! I was mainly confused about this one above but it seems like its the same odd function but with just the minus on the other side.
So basically:
The even function: even if you flip the sign of the input, the output is still the same.
Odd function: if you flip the sign of the input, its the same output but negated.
Is this correct?
Uh yeah