Finding The Domain Without The Graph - Functions - Cool Math

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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

The domain is = ( -infinity , 3 / 2 ) U ( 3 / 2 , infinity )

TRY IT:

Find the domain of f( x ) = 6 / ( 5x + 3 ) *show work!!

 

How about this one?

f( x ) = square root( x + 5 )

Square roots -- what do we know about square roots?

                    square root( 16 ) = 4  ...  So, 16 is OK to put in.

                    square root( 0 ) = 0  ...  So, 0 is OK.

                    square root( 3.2 ) is about 1.788  ...  Yuck!  But, 3.2 is OK.

                    square root( -25 ) = ?  ...  Nope!  Can't do it!                                                                          *We only want real numbers!

No negatives are OK!

square root( inside )

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  inside is greater than or equal to 0  and solve it!

Now, let's find the domain of

f( x ) = square root( x + 5 ) ... x + 5 is greater than or equal to 0 ... x is greater than or equal to -5

So, the domain of f( x ) = square root( x + 5 ) is [ -5 , infinity ) .

TRY IT:

Find the domain of f( x ) = square root( 3 - x ) . *Show work!!

Here's a messier one:

Let's find the domain of f( x ) = square root( 7 - 8x )
Set  7 - 8x is greater than or equal to 0  and solve!

7 - 8x is greater than or equal to 0 ... subtract 7 from both sides ... -8x is greater than or equal to -7 ... divide both sides by -8 ... x is less than or equal to 7 / 8

The domain is ( -infinity , 7 / 8 ] .

YOUR TURN:

Find the domain of f( x ) = square root( 4x - 5 ) .  *Show work!

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