Domain of function

[dcowboys107]

(1-x^2)^(1/2)

--------------

(4-x^2)^(1/2)

 

 

One minus x squared divided by 4 minus x squared all under a square root sign.

 

My teacher says the domain is (-inf,-2) union [-1,1] union (2,inf)

 

Is it necessary to include the [-1,1]?

 

Wouldn't it be obvious that function is valid at that point?

 

I have a tough time deciding when to include those terms

 

How do I know when to include the infinity and such?

[cypress]

(1-x^2)^(1/2)

--------------

(4-x^2)^(1/2)

 

 

One minus x squared divided by 4 minus x squared all under a square root sign.

 

My teacher says the domain is (-inf,-2) union [-1,1] union (2,inf)

 

OK that looks correct to me.

 

Is it necessary to include the [-1,1]?

 

Wouldn't it be obvious that function is valid at that point?

 

You have to include it if you want the answer to be complete and correct. I might say it is all obvious so n answer is required but I think I might get the problem marked wrong if I were to say that.

 

I have a tough time deciding when to include those terms

 

How do I know when to include the infinity and such?

 

The trick is to first identify where the function is undefined.

 

1) You know it is undefined when the denominator is 0

2) You also know that negative square roots are undefined or at least not real... and so I am assuming the problem asks for real values.

 

then once you have the parts where it is undefined you include everything that is defined.

 

I like to do it this way:

 

for 1)

it is undefined at -2 and 2

 

for 2)

the numerator is negative from -inf to -1 and from +1 to +inf

but the denominator is negative from -inf to -2 and from +2 to inf

 

So including both the numerator and denominator it is undefined between -2 and -1 and then +1 and +2 inclusive

 

It helps to draw this on a line graph.

 

Then you put this information together to get the answer your teacher provided. I find it helps to go one step at a time and write everything down.