Question

I just cant seem to get my head around what it does to the functions, for example the domain.

Eg f(x) = |ln(x)| or ln|x|

What does it do to the function? and how does it affect the domain?

also is tan x a function of ALL real numbers?

Answer 1

im talking about the absolute value in the first part, what does | | do to the functions

Answer 2

tan(x) has a restricted domain..

Answer 3

thats what I thought, thanks. How about the other part please

Answer 4

taking the absolute value of the function after it is evaluated just makes the answer positive.
e.g. ln(.5)=-0.693..., abs(ln(.5))=0.693...\[ \ln(.5)=-0.693..., \left| \ln(.5) \right| =0.693...\]
putting the absolute value inside the log function restricts the domain to only positive numbers, since the log of a negative number is imaginary.

Answer 5

e.g. \[\frac{\ln(-1)}{\sqrt{-1}}=\pi\]

Answer 6

so If i get a question like say, what is the domain of | ln (x) |

the domain is just the usual x > 0 ? because any negative answers will be turned to positive?

Answer 7

yeah, taking the absolute value of a function changes the range, not the domain.

Answer 8

ah I see, ok I get it now. One more question please, what if im taking the limit of

a absolute function?

Answer 9

You mean\[\lim_{x \rightarrow a}\left| x \right|?\]

Answer 10

yup

Answer 11

or

lim | x^2 + 2x + 6|
--------------
2x
x -> a

Answer 12

Evaluate as normal, I suppose. I'd want to doublecheck this, but I think\[\lim_{x \rightarrow a}\left| x \right|=\left| \lim_{x \rightarrow a}x \right|.\]

Answer 13

In that case, where only the numerator is in absolute value, then I think you might have to check where the numerator goes to zero or negative just in case. It'll depend on what a is. If a=1, then nothing funny is going on.

Answer 14

so basically the limit of a absolute value function if negative, will be turned to positive?

Answer 15

In the example you gave, if x goes to a particular negative number, then the limit will be negative because you're only taking the absolute value of the numerator.
In that example, unless x approaches 0, you can just evaluate as normal and the limit is equal to the function value.

Answer 16

Oh, nevermind, I checked the specific function and the numerator is never negative, so the limit is only negative if x is negative, but then since the numerator is never negative, the absolute value function has nothing to do.

Answer 17

Ok i see, but generally if I have a function in which the whole part is inside the abs lines, the limit if negative shall be turned to positive?

Answer 18

yes, because you're taking the limit of a guaranteed positive number.

Answer 19

thank you so much, that helps a lot!

Answer 20

Let me know if you come across any really weird examples. I hardly ever see absolute value functions show up while doing calculus.

Answer 21

yup, just going over some past exam papers. Will ask thanks!.