Question 62555
Find the Domain and Range of the function: 1/x<sup>4</sup>.
<pre><font size = 5 color = "darkblue"><b>
There are only two things which restrict the domain 
or range of a function in ordinary algebra.

1.  Denominators which must never be 0.
2.  Even root radicands which must never be negative.

If there are no denominators or even root radicands 
which contain variables then the domain is always
(-<font face = "symbol">¥</font>, <font face = "symbol">¥</font>)

             1
y = f(x) = ----
            x<sup>4</sup>

has a denominator with a variable, so we must 
require that x<sup>4</sup> <font face = "symbol">¹</font> 0, or x <font face = "symbol">¹</font> 0

So the domain is 

      (-<font face = "symbol">¥</font>, 0) <font face = "symbol">È</font> (0, <font face = "symbol">¥</font>)

To find the range, we solve the equation for x, 
and use the same criteria for y.

             1
       y = ----
            x<sup>4</sup>

Multiply both sides by x<sup>4</sup>

     x<sup>4</sup>y = 1

            1 
      x<sup>4</sup> = ---
            y

             1
      x =  ---<u>-</u>-
            <sup>4</sup><font face = "symbol">Ö</font>y

This is an even root radical, therefore its radicand, y
must not be negative. Since it is also in a denominator
it must not be 0 either.  Therefore it must by greater
than 0, or y > 0.  Therefore the range is

       (0, <font face = "symbol">¥</font>)

Edwin</pre>