f(x)= 9x³+4x²-7

The DOMAIN is the set of all numbers on the x-axis, such that if
you were to draw a VERTICAL line through it in both directions up
and down long enough it would intersect the curve.


The RANGE is the set of all numbers on the y-axis, such that if
you were to draw a HORIZONTAL line through it in both directions right
and left long enough it would intersect the curve.



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Notice 

1. That the graph is such that if you were to draw a VERTICAL line through 
any point on the x-axis in both directions long enough it would intersect the
curve.

2. That the graph is such that if you were to draw a HORIZONTAL line through 
any point on the y-axis in both directions long enough it would intersect the
curve.

That's why both the domain and the range of f(x) are (-infinity,infinity)

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Another way to look at it is:

The DOMAIN of y = 9x³+4x²-7 consists of all the numbers you can substitute 
for x and get a value for y.  The expression 9x³+4x²-7 contains no
denominators that could ever be 0 and there are no square roots that
could have negative numbers under them.  

Therefore there are no restrictions on what numbers you can substitute for
x and get a value for y.

So the domain is "ALL REAL NUMBERS" or (-infinity, infinity) 


The RANGE of y = 9x³+4x²-7 consists of all the numbers you can GET 
for y by substituting value for x.  The expression 9x³+4x²-7 contains no
denominators that could ever be 0 and there are no square roots that
could have negative numbers under them.  

 

Notice that it goes infinitely low on the left and infinitely high on the
right.  And there are no gaps in the curve anywhere.

So the RANGE is also "ALL REAL NUMBERS" or (-infinity, infinity)  

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The DOMAIN for every ODD-DEGREE polynomial function is (-infinity, infinity).

The RANGE for every ODD-DEGREE polynomial function is (-infinity, infinity).

The DOMAIN for every EVEN-DEGREE polynomial function is (-infinity, infinity).

However the RANGE for an EVEN-DEGREE polynomial is NOT (-infinity, infinity).

The range of an EVEN-DEGREE polynomial is either:

(-infinity, MAXIMUM VALUE OF y) or  (MINIMUM VALUE OF y, infinity).

Edwin