Finding the Domain of a Function
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 |
Do you see any x guys that would cause a problem here?
What about | ? |
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:
Do the interval notation in two pieces:
domain |
YOUR TURN:
Find the domain of |
Sometimes, you can’t find the domain with a quick look.
Check it out:
Let’s find the domain of |
Hmm… It’s not so obvious!
BUT, we are still looking for the same thing:
The bad x that makes the denominator 0! |
How do we find it? Easy!
Set the denominator = 0 and solve!
The domain is |
TRY IT:
Find the domain of | *show work!! |
How about this one?
Square roots — what do we know about square roots?
… So, 16 is OK to put in.
… So, 0 is OK.
… Yuck! But, 3.2 is OK.
… Nope! Can’t do it!
*We only want real numbers!
No negatives are OK!
The inside of a radical cannot be negative if we want real answers only (no i guys). So, the inside of a radical has to be 0 or a positive number.
Set | and solve it! |
Now, let’s find the domain of
So, the domain of | is | . |
TRY IT:
Find the domain of | . *Show work!! |
Here’s a messier one:
The domain is | . |
YOUR TURN:
Find the domain of | . *Show work! |