Question 1196851
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Clearly x = -1 is a critical value in determining the domain and range.<br>
At x = -1, the function is<br>
{{{(-1+1)^(-1-1)=0^(-2)=1/0^2}}}<br>
which is undefined.<br>
So x = -1 is not in the domain.<br>
For any value of x greater than -1, the expression is a power of a positive number, so the function value can always be evaluated.  So (-1, infinity) is at least part of the domain.<br>
The tricky part of determining the domain is with values of x less than -1.  That results in a negative number raise to a negative power.<br>
A negative number raised to a negative power is a real number if and only if the power is an integer.  For example...<br>
{{{f(-2)=(-1)^(-3) = 1/(-1)^3 = -1}}}
{{{f(-3)=(-2)^(-4)=1/(-2)^4=1/16 = 0.625}}}
{{{f(-4)=(-3)^(-5)=1/(-3)^5=-1/243=-0.0041}}}<br>
But for non-integer values of x less than -1, the function value is not a real number.<br>
So....<br>
Depending on your definition of domain, the domain of the function is either<br>
(-1,infinity)<br>
or<br>
(-1,infinity) U {integers less than -1}<br>
For the range, unless you know some very high-level mathematics, you can use a graphing calculator to find the answer.  It should be clear that there is no upper limit to the range; my TI-83 calculator shows a minimum value of approximately 0.815154 at approximately x=0.454731.<br>
So the range is approximately (0.815154,infinity).<br>