Yahoo Answers: Answers and Comments for Find the domain and range of f(x)=ln (3+sin x)? [Mathematics]
Copyright © Yahoo! Inc. All rights reserved.
https://answers.yahoo.com/question/index?qid=20100905154346AAFood3
From s93z
enUS
Sun, 05 Sep 2010 15:43:46 +0000
3
Yahoo Answers: Answers and Comments for Find the domain and range of f(x)=ln (3+sin x)? [Mathematics]
292
38
https://answers.yahoo.com/question/index?qid=20100905154346AAFood3
https://s.yimg.com/zz/combo?images/emaillogous.png

From No Mythology: The domain of the sine function is all of the ...
https://answers.yahoo.com/question/index?qid=20100905154346AAFood3
https://answers.yahoo.com/question/index?qid=20100905154346AAFood3
Sun, 05 Sep 2010 18:39:39 +0000
The domain of the sine function is all of the reals. With the natural log, you have to restrict x so that the argument of the log is positive. But note that for all x, 1 ≤ sin(x) ≤ 1. So inside the log
2 ≤ 3 + sin(x) ≤ 4.
That is, 3 + sin(x) is always positive. The domain of f is therefore all real numbers
domain = (∞, ∞)
As for the range, the above shows that the term inside of the log can take on every value from 2 to 4 including the end points. So the range is all values from ln(2) to ln(4).
range = [ln(2), ln(4)]
The function is periodic with period 2π because the sine is periodic with period 2π. The sine function has the property
sin(x + 2π) = sin(x) for all real x.
And 2π is the smallest value for which this is true. Then we see that
f(x + 2π) = ln(3 + sin(x + 2π)) = ln(3 + sin(x)) = f(x).
f is neither even nor odd. The sine function itself is odd, but the constant function 3 is even. The sum 3 + sin(x) is neither. The log by itself has no symmetry, though that doesn't preclude it being involved in a symmetric function. Note that
f(x) = ln(3 + sin(x)) = ln(3  sin(x))
which is not the same as f(x) or  f(x).