How do you find the range of a function

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 ]  .

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c) they were beaten and fired from their jobs.

the first action that caused many creek indians to turn against william mcintosh was when mcintosh a) supported the united states during the creek war of 1813–1814. b) attempted to stop the sale of creek land to the u.s. government. c) seized control of over 1,000 acres of land belonging to the creeks.