Expert: Scotto - 7/15/2009

Question
I have two problems.
1) f(x)=x*sqrt(7-x^2)

2) g(x)= sin(x)/1+cos(x)

How would you find the domain?
I know you are suppose to find out if any numbers that won't work. but i don't know where to begin.

Answer
The domain is all of the x that can be put into the function.

1) This function has an x and a sqrt(7-x^2).
The x can have any value put into it.

The sqrt(7-x^2) is restricted in that 7-x^2 must never be negative.
That means that x^2 <= 7.
That means that -sqrt(7) <= x <= sqrt(7).

Therefore since the x can have any x, that is the total domain of the function.

The best way to sketh it is to use Excel.

Put -2.5 in A1.  Put =A1*sqrt(7-A1^2) in B1.

Put =A1+.5 in A2.  It should make it =A1+0.5. Copy B1 to B2.

Highlight A2 thru B11.

Using the menu, do an Edit-Fill-Down.

Here are the results:
-2.5   -2.165063509
-2   -3.464101615
-1.5   -3.269174208
-1   -2.449489743
-0.5   -1.299038106
0   0
0.5   1.299038106
1   2.449489743
1.5   3.269174208
2   3.464101615
2.5   2.165063509

It looks like the function has a minimum somewhere around -2 and a maximum somewhere around 2.


2) For sin(x), any value can be put in.
For 1+cos(x), the only value that can't be put in is where
1+cos(x)=0.  For this to be true, we would have to have -1 = cos(x).
The first place this occurs is at pi.  Since the function repeats every 2pi, all of the places would be pi +/- 2n*pi for any integer (positive or negative) n.

The domain would be all of the x except for the set of points
pi +/- 2n*pi { n=integer }.

This one could also be graphed in Excel.
I would put 0 in A1.  In B1, put =sin(A1)/(1+cos(A1)
In A2, put =A1+pi()/5.  In B2, copy B1.

Highlight A2:B11 and Fill-Down.  In B6, it will say #DIV0!
This means you are trying to divide by 0, which is to be expected.
It can be seen that when X is less than PI, the function value is positive.  When X is greater than PI, the funciton value is positive.
This will repeat every 2pi.

It should be noted if x = 2npi, where n is any integer, the function has the value 0.  In that way, it resembles a cubic, except this function has an assymptote at every npi where n is an odd number.

Here are the results from Excel:
0   0
0.628318531   0.324919696
1.256637061   0.726542528
1.884955592   1.37638192
2.513274123   3.077683537
3.141592654   #DIV/0!
3.769911184   -3.077683537
4.398229715   -1.37638192
5.026548246   -0.726542528
5.654866776   -0.324919696
6.283185307   -1.22515E-16