What is the domain and range of #y= (4x^2 - 9) / ((2x+3)(x+1))#?

Answer 1

See below.

Notice:

#4x^2-9# is the difference of two squares. This can be expressed as:
#4x^2-9=(2x+3)(2x-3)#

Changing this in the numerator:

#((2x+3)(2x-3))/((2x+3)(x+1))#

Removing similar factors:

#(cancel((2x+3))(2x-3))/(cancel((2x+3))(x+1))=(2x-3)/(x+1)#
We notice that for #x=-1# the denominator is zero. This is undefined, so our domain will be all real numbers #bbx# #x!=-1#

This can be stated as follows in set notation:

#{x in RR | x != -1}#

or using interval notation:

#(-oo , -1)uu(-1,oo)#

To determine the range:

We know the function is undefined for #x=-1#, therefore the line #x=-1# is a vertical asymptote. The function will go to #+-oo# at this line.
We now see what happens as #x ->+-oo#
Divide #(2x-3)/(x+1)# by #x#
#((2x)/x-3/x)/(x/x+1/x)=(2-3/x)/(1+1/x)#
as: #x->+-oo# # \ \ \ \ \ (2-3/x)/(1+1/x)=(2-0)/(1+0)=2#
This shows the line #y=2# is a horizontal asymptote. The function can't therefore ever equal 2.

Thus, the range can be stated as follows:

#{y in RR| y !=2}#

or

#(-oo,2)uu(2 , oo)#

This is evident from the function's graph:

graph{(2x-3)/(x+1) [-16.23, 16.25, -32.48, 32.44]}

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Answer 2

The domain of the function ( y = \frac{{4x^2 - 9}}{{(2x + 3)(x + 1)}} ) is all real numbers except ( x = -1 ) and ( x = -\frac{3}{2} ). The range is all real numbers except ( y = 4 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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