Question 215432
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You are spot on, the domain is *[tex \Large \{\,x\,|\,x\ \in\ \R,\ x \neq\ \pm 4\}]


The range is the set of values that the function can assume for all possible values of the independent variable in the domain.


Let's look at values of *[tex \Large x] in the interval *[tex \Large -4\ <\ x\ <\ 4].  The square of any value in this interval is going to be smaller than 16, so the value of the function on this interval will be negative or zero.  Hence, on this interval, the maximum value of the function is zero, and there is no minimum value of the function because you can continue to choose values of *[tex \Large x] closer and closer to either 4 or -4 making the function get closer and closer to *[tex \Large -\infty].


Values of the independent variable either slightly greater than 4 or slightly less than -4 will result in a very large positive value for the function.  Again, you can make the function as large as you want by selecting a value as close as you want to either -4 or 4.


As the absolute value of the independent variable gets very large, that is as you approach either *[tex \Large -\infty] or *[tex \Large \infty], the constant term in the denominator has less and less effect on the overall value of the quotient.  For example, if *[tex \Large x = 5] you have *[tex \Large \frac{25}{9}], a little less than 3.  But if *[tex \Large x = 100], you have *[tex \Large \frac{10000}{9984}] a number much closer to 1.  In fact, 1 is the limiting number.  For any value *[tex \Large >4] or *[tex \Large <-4], the function will always be *[tex \Large > 1]


Taking it all together, you can see that the range excludes anything in the interval *[tex \Large 0\ \leq\ x\ <\ 1], hence the range is defined as


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \{\,x\,|\,x\ \in\ \R,\ x\ \leq\ 0\}\ \small{\cup}\LARGE\ \{\,x\,|\,x\ \in\ \R,\ x\ >\ 1\}



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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