SOLUTION: 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. How to solve?

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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.
How to solve?
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
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.

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

g%28x%5E%22%22%29=x%5E2-3
g%28h%28x%29%5E%22%22%29+=+%28h%28x%29%5E%22%22%29%5E2-3
g%28sqrt%28x-6%29%5E%22%22%29+=+%28%28sqrt%28x-6%29%29%5E%22%22%29%5E2-3
goh+=+%28x-6%29-3
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%282%2C1%2Cx-6%3E=0%2Cx%3E=6%29

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

which in interval notation is

matrix%281%2C5%2C%22%5B%22%2C6%2C%22%2C%22%2Cinfinity%2C%22%29%22%29

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

%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2Cinfinity%29%29
 
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%281%2C5%2C%22%5B%22%2C6%2C%22%2C%22%2Cinfinity%2C%22%29%22%29


Edwin