
Domain and Range
Hi, just wonderin if I could have some help with these two questions: each of them asks to find the domain and the range using calculus techniques, and as I am reveiwing this after doing this 12 weeks ago, I am a bit hazy as to what's required.
(a)f(x)=cos sqrt(e^x1)
(b) g(x)=1//(x^2+1)
if anyone could help and explain fully, that would be greatly appreciated :)

Re: Domain and Range
(a) domain ...
(why?)
range ...
is between what two values?
(b) domain ...
for all , so what is the domain?
range ...

Re: Domain and Range
Basically finding the domain means that all the places in where the function is defined. So for example
1) Find the domain of . The domain is All real numbers except 1. Because and u cannot divide by 0.
So for ur first example
1) . cos(x) is defined for all real numbers, but is real only for (because is not a real number, its a complex number).
As for the range, range is basically all the values ur function can take if u plug in x's from ur domain only.
For example
2) Find the Range of . Since we said the domain of f was all real numbers except 1. We see that pluggin in , for makes the denominator in makes the denominator < 1, which means the function it it self can become as large as So for our range is . Now take So our range can take on values from So our range for x > 1 is plus which means range is Now since we did x > 1, now we try x < 1, so for x < 1, we see that for if our denominator takes on the values in the interval . and if x < 0, we get a small negative denominator, and the function takes on every value from [1, 0). So our range is , which means that our whole range is , basically ur function can take on every value except 0.