Melody

HiNTS:

It is increasing when the gradient of the tangent is positive    i.e.       \(\frac{dy}{dx}>0\)

It is decreasing when the gradient of the tangent is negative    i.e.       \(\frac{dy}{dx}<0\)

It is even if it is symmetrical around the y axis, which is the line x=0    SO      \(f(-x)=f(x)\)

It is odd if it is has 180 degree rotational symmetrical around the (0,0), SO      \(f(-x)=-f(x)\)

It is invertible is both the original and the inverse are functions. This happens when:

It is invertible is the gradient is always increaing or always decreasing, 0 is ok too. 

     \(\frac{dy}{dx}\ge0\quad or \quad\frac{dy}{dx}\le0 \qquad \text{ for all real values of x}\)  

Now you can do them yourself :)

\(\frac{1}{\sqrt3}n=\frac{n}{\sqrt3}=\frac{n}{\sqrt3}\times \frac{\sqrt3}{\sqrt3}=\frac{\sqrt3\;n}{3}\)

I just deleted a couple of expletives .   I didn't change the meaning in any way.

OAlso this is guests post and opinion only. It does not necessarily reflect any one elses view.

LOL

you didn't copy it properly. The \t   that should have been in front of  "ext... "  is missing and that is why the sentence is not displying properly.

I look forward to you being an account holder :)

Thanks asdf :)

Invertible.

I expect it means more than that.

It probably means every x has just one y  AND every y has just one x.

So that it is a function for all values of x and its inverse is also a function for all values of x.

I quickly looked it up. That seems to be what it means.

Maybe guest put it there by mistake. Anyway, it is gone now :)

What does 'invertible' mean?

increasing is when the gradient of the tangent is positive

decreasing is when the gradient of the tangent is negative

If $|z| = 4$ and $|w| = 2$, then what is the imaginary part of $\dfrac{z}{w}$?​

\(w=2e^{i \theta}\\ z=4e^{ i(\theta+\frac{\pi}{4} )}\\~\\ \frac{z}{w}=\frac{4e^{ i(\theta+\frac{\pi}{4} )}}{2e^{i \theta}}\\ =2e^{i\theta+\frac{\pi i}{4}-i\theta}\\ =2e^{\frac{\pi i}{4}}\\ =2cos\frac{\pi}{4}+(2sin\frac{\pi}{4})i\\ =\frac{2}{\sqrt2}+\frac{2}{\sqrt2}i\\ =\sqrt2+\sqrt2i\)

So the imaginary part is    \(\sqrt2\cdot i\)

Thanks KillerSteve. :)