OK, so suppose we don't have the graph of a function to look at like in the last section...
Can we still find the domain and range?
Domains: Yes (as long as the algebra doesn't
get too hairy... and it won't for us.)
Ranges: Not really (you usually need the
picture -- unless it's something
really basic.)
So, we'll just be doing domains on these -- which is really where the action is anyway.
Asking for the domain of a function is the same as asking
"What are all the possible x guys
that I can stick into this thing?"
Sometimes, what you'll really be looking for is
"Is there anything I CAN'T stick in?"
Check it out:
Let's find the domain of f( x ) = 2 / ( x - 3 )
Do you see any x guys that would cause a problem here?
What about x = 3 ?
f( 3 ) = 2 / ( 3 - 3 ) = 2 / 0 ... ouch!
So, x = 3 is a bad guy! Everyone else is OK, though.
The domain is all real numbers except 3.
What would the interval notation be?
When in doubt, graph it on a number line:
number line showing the domain is all numbers except 3
Do the interval notation in two pieces:
domain = ( -infinity , 3 ) U ( 3 , infinity )
YOUR TURN:
Find the domain of f( x ) = 5 / ( x + 7 )
Sometimes, you can't find the domain with a quick look.
Check it out:
Let's find the domain of f( x ) = 1 / ( 3 - 2x )
Hmm... It's not so obvious!
BUT, we are still looking for the same thing:
f( x ) = 1 / ( 3 - 2x ) The bad x that makes
the denominator 0!
How do we find it? Easy!
Set the denominator = 0 and solve!
3 - 2x = 0 ... subtract 3 from both sides ... -2x = -3 ... x = -3 / -2 = 3 / 2
