[Roxy]

How do I find
1. Domain & Range
2. Intervals of increase/decrease
3. max./min values
4. intercepts
5. asymptotes
6. concavity

for Exponential and Logarithmic functions. Can someone explain how I do this in detail plzzz

These are the types of questions I have (plz don't solve them just tell me how to do them:

y= 2x - lnx

y = x^2lnx

y= in(x-1)^2

y = e^x + 1

y= x - lnx

[CroCivic91]

English is not my first language so I may say some words wrong, but I'll try to explain it in detail.

So, a domain of a function f(x) is a set of elements (in this example, a set of numbers x) for which you can compute f(x). So for example, a domain for f(x)=1/x is the whole set R without 0, because you cannot compute 1/0. Now for exponential function, it's domain is the whole set R, because which ever number (x) from R you take, you can compute e to the power of x. e to the power of 0 is 1, e to the power of -1 is 1/e and so on. Now to fully understand logharitmic function, you must notice that it is an inverse function of exponential function. And that the image (a set of numbers that you can "hit" when calculating e to the power of x (in mathematical terms - Image(f) = { f(x) | x is an element of domain of f }) ) of exponential function is the domain of logharitmic function. Now you can see the Image(e to the power of x) = { x | x > 0 } (a set of all numbers strictly larger than zero). That means that you can compute log(x) only for such x which is strictly larger than zero. You cannot compute log(0), nor log(-1). But log(10)=1.

Now with that in mind, you know domain of log(x) is a set of all numbers strictly larger than zero.
Now for example: lets say you have f(x) = log( x - 1 )
Now to find out what it's domain is, you can use the above mentioned rule: it's parameter must be strictly larger than zero.
That means: x - 1 > 0 and that means that x must be strictly larger than 1 for you to be able to calculate f(x). If you try to calculate f(1) you will get f(1)=log( 1 - 1 ) = log(0) which you can't calculate.

Now another issue is if you have something like
f(x) = log(x) + square_root(x)
Now you can calculate square_root(x) for such x which is larger than zero, or even zero itself. But you cannot calculate log(x) for x == 0, which means that domain of f(x) is an intersection of all domains of it's "subfunctions". In general: lets say that f(x) = f1(x) + f2(x) * f3(x)
Then Domain( f ) = Domain( f1 ) intersected with Domain( f2 ) intersected with Domain( f3 )

Ok, that should explain finding out the domain.

Now to find out min/max values, you have to use some mathematical theorems, and you must know how to calculate a derivative of a given function. One theorem states: If a function has a minimum/maximum in point m, then f'(m) == 0. Notice that if f'(m) == 0, it doesn't strictly mean that m is a minimum/maximum of a function.

Confusing, eh? Let's see some examples: lets find out a min/max of f(x) = x^2 + 2x + 1
Now find the derivative (I hope that's what you call that thing in english)
f'(x) = 2x + 2
now f'(x) = 2x + 2 == 0 if x == -1
Now if you were to draw a graph of f(x), you would find out that it has a minimum in x == -1.
For another example, f(x) == x^3.
f'(x) = 3x^2 == 0 if x == 0, but if you were to draw a graph of x^3 you would find that it has no min/max in x == 0.

Now you call those points "stationary points". You find out if they are min/max/nothing with second derivative.

Ok I just found out I am missing too many words to explain why this is like that, and it would take like an hour to explain.

On to intervals. Basically, if you know your function is continuous (it's graph doesn't "jump") between stationary points x1 and x2, then you just calculate first derivative in a freely chosen point between x1 and x2 (you can take a point in half the distance between those two) and if it's larger than zero, the function increases. If it's lower than zero, the function decreases.

Ok, sorry, I got to go to class right now. Will try to explain the rest later :)

Hope you find this at least a bit useful...

Kresho

[Roxy]

I'm confused... :confused: with all the explaining.

You can just say stuff like set y' = 0 and stuff like that for all of them.

And thanks for trying to help