<PRE><B>Question 49755</B>
&lt;pre&gt;&lt;b&gt;&lt;font size=4&gt;if f(x)=3x+1,and g(x)=(x&lt;sup&gt;2&lt;/sup&gt;+5x)&lt;sup&gt;-1/2&lt;/sup&gt;,find g(f(x))

g(A) means to plug in A for x in the right side of g(x) and simplify.

So,

g(f(x)) means to plug in f(x) for x in the right side of g(x) and 
simplify.

But since f(x) = 3x+1

g(f(x)) means to plug in 3x+1 for x in the right side of g(x) and 
simplify. 

So we want to take out all the x&#39;s in this:

g(x)=(x&lt;sup&gt;2&lt;/sup&gt; + 5x)&lt;sup&gt;-1/2&lt;/sup&gt;

putting blanks in place of all the x&#39;s:

g(____) = (______&lt;sup&gt;2&lt;/sup&gt; + 5______)&lt;sup&gt;-1/2&lt;/sup&gt;

I will change the parentheses on the right to brackets to avoid
parentheses within parentheses, since parentheses must go in the 
blanks on the right.  The left already has parentheses.

g(____) = [______&lt;sup&gt;2&lt;/sup&gt; + 5______]&lt;sup&gt;-1/2&lt;/sup&gt;

Now fill in all the blanks where x was with f(x) written as what 
it is given equal to, namely (3x+1) 

So g(f(x)) =

g(&lt;u&gt;3x+1&lt;/u&gt;) = [&lt;u&gt;(3x+1)&lt;/u&gt;&lt;sup&gt;2&lt;/sup&gt; + 5&lt;u&gt;(3x+1)&lt;/u&gt;]&lt;sup&gt;-1/2&lt;/sup&gt;

g(3x+1) = [(3x+1)&lt;sup&gt;2&lt;/sup&gt; + 5(3x+1)]&lt;sup&gt;-1/2&lt;/sup&gt;

        = [(3x+1)(3x+1) + 5(3x+1)]&lt;sup&gt;-1/2&lt;/sup&gt;

        = [9x&lt;sup&gt;2&lt;/sup&gt;+6x+1 + 15x+5]&lt;sup&gt;-1/2&lt;/sup&gt;

        = (9x&lt;sup&gt;2&lt;/sup&gt;+21x+6)&lt;sup&gt;-1/2&lt;/sup&gt;

Edwin&lt;/pre&gt;&lt;/font&gt;&lt;/b&gt;</PRE>