Respuesta :
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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To find the excluded value in the domain of the function, equate the denominator to zero and solve for . So, the domain of the function is set of real numbers except . The range of the function is same as the domain of the inverse function. So, to find the range define the inverse of the function.
