<PRE><B>Question 1075749</B>
&lt;pre&gt;&lt;font size = 5&gt;&lt;b&gt;
&lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; = sin&lt;sup&gt;-1&lt;/sup&gt;(x)+cos&lt;sup&gt;-1&lt;/sup&gt;(x)-tan&lt;sup&gt;-1&lt;/sup&gt;(x) for the domain x &amp;#8805; 0,
what is the range?

Because of the inverse sine and cosine functions, 
the domain of &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; is 0 &amp;#8804; x &amp;#8804; 1, 

We find the derivative of &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt;

{{{d(theta)/dx}}}{{{&quot;&quot;=&quot;&quot;}}}{{{1/sqrt(1-x^2)-1/sqrt(1-x^2)-1/(1+x^2)}}}

The first two terms cancel and we have

{{{d(theta)/dx}}}{{{&quot;&quot;=&quot;&quot;}}}{{{-1/(1+x^2)}}}

Since the derivative of &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; is negative, the function &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt;
decreases everywhere on the open part of its domain,
which is 0 &lt; x &lt; 1.
[&lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; is not differentiable at the endpoints 0 and 1, although 
&lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; is defined at those endpoints]

We evaluate &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; at the endpoints of the 
domain of &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; : 

Substitute x = 0

&lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; = sin&lt;sup&gt;-1&lt;/sup&gt;(0) + cos&lt;sup&gt;-1&lt;/sup&gt;(0) - tan&lt;sup&gt;-1&lt;/sup&gt;(0) = 0° + 90° - 0° = 90°

Substitute x = 1

&lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; = sin&lt;sup&gt;-1&lt;/sup&gt;(1) + cos&lt;sup&gt;-1&lt;/sup&gt;(1) - tan&lt;sup&gt;-1&lt;/sup&gt;(1) = 90° + 0° - 45° = 45°

Thus the range for &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; for the domain
 
0 &amp;#8804; x &amp;#8804; 1

is

45° &amp;#8804; &lt;font face=&quot;symbol&quot;&gt;q&lt;/font&gt; &amp;#8804; 90°

Edwin&lt;/pre&gt;&lt;/font&gt;&lt;/b&gt;

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