Question 1151905
For the real-valued functions = g(x)=x^2-3 and h(x) = sqrt(x-6), find the
composition g∘h and specify its domain using interval notation.
<pre>

Substitute the right side of h(x) for x in the equation for g(x)

{{{g(x^"")=x^2-3}}}
{{{g(h(x)^"") = (h(x)^"")^2-3}}}
{{{g(sqrt(x-6)^"") = ((sqrt(x-6))^"")^2-3}}}
{{{goh = (x-6)-3}}}
{{{goh = x-6-3}}}
{{{goh = x-9}}}

Since g∘h = g(h(x)), we cannot substitute anything for x unless it is in the
domain of h, and also what we substitute for x must not cause h(x) to produce
any values which are not in the domain of g.

We find the domain of h:
{{{matrix(2,1,x-6>=0,x>=6)}}}

----------------------------☻======>
-3 -2 -1  0  1  2  3  4  5  6  7  8

which in interval notation is

{{{matrix(1,5,"[",6,",",infinity,")")}}}

The domain of g, since g is a polynomial function, is "all real numbers",
which is written

{{{(matrix(1,3,-infinity,",",infinity))}}}
 
Thus h cannot produce any value which is not in the domain of g, so the
domain of g∘h is the same as the domain of h

{{{matrix(1,5,"[",6,",",infinity,")")}}}


Edwin</pre>