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find the domain and range

suppose that f(x) = √(x2 -72)  and g(x)= √(7-x)
 
for each function (h) below, find a formula for h (x) and the domain of h ( use internal notation)
a) h(x) = f composed with g of (x)
b) h(x) = g composed with f of (x)
c) h(x)= f composed with f of (x)
d) h(x)= g composed with g of (x) 
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1 Answer

For f(x)=√(x²-7²) and g(x)= √(7-x)
a)  h(x)=f(g(x))=√(g²(x)-7²)=√(7-x-7²)=√(7-x-49)=√(-x-42)
The domain are those x-values where -x-42≥0 or where x+42≥0 or where x≥-42
For these values h(x)=√(-x-42) takes values in the range [0,∞), or all positive values.
 
b)  h(x)=g(f(x))=√(7-f(x))=√(7-√(x²-7²))=√(7-√(x²-49))
In order to determine the domain we must be able to take square roots so
1.  x²-49 must be ≥0 or x²≥49 or either x≤-7 or x≥7 an in addition
√(x²-49) must be ≤7 since we need √(7-√(x²-49)).
√(x²-49)≤7 means that  x²-49≤49 or that x²≤98 or that x≤√98=√(2×49)=7√2 or x≥-7√2
Combining the requirements the domain is the two intervals   -7√2≤x≤-7  or 7≤x≤7√2
The range of h(x) is h(7) to h(7√2) or from 0 to √7.  The function has the same range when the negative boundaries of the domain are used.
 
c) f(f(x))=√(f²(x)-49)=√(x²-49-49)=√(x²-98).  The domain are those values where x²-98≥0 or where
x²≥98 or where x≥√98 or where x≤-√98, That is the two infinite intervals x≤-7√2 and x≥7√2
The range of the function is all the positive numbers.
 
d) g(g(x))=√(7-g(x))=√(7-√(7-x)).  In order to evaluate the square roots we need to have x≤7 for √(7-x) and 7≥√(7-x) for the final square root.  This last means that 49≥7-x or that x≥-42.  The domain is
-42≤x≤7.  The range is from 0 to √7