Determine the functions f∘g,g∘f,f∘f and g∘g and their domains if f(x)=1√x and g(x)=x2−4x
For f∘g,
f∘g=f(g(x))Definition of f∘gf∘g=1√x2−4xDefinition of f
Since the function involves square root in the denominator, we want
x2−4x>0x(x−4)>0
The factors on the left hand side are x and x−4. These factors are zero when x is and 4, respectively. These numbers divide the number line into interval
(−∞,0),(0,4),(4,∞)
By testing some points on the interval...
Thus, the domain where x(x−4)>0 is (−∞,∞)⋃(4,∞)
For g∘f,
g∘f=g(g(x))Definition of g∘fg∘f=(1√x)2−4(1√x)Definition of fg∘f=1x−4√xDefinition of g
Since the function is a rational function that invovles square root, so the domain of g∘f is (0,∞)
For f∘f,
f∘f=f(f(x))Definition of f∘ff∘f=1√1√xDefinition of ff∘f=1√1√xSimplifyf∘f=4√x
We know that if the index is any even number, the radicand can't have a negative value. So the domain if f∘f is [0,∞)
For g∘g,
g∘g=g(g(x))Definition of g∘gg∘g=(x2−4x)2−4(x2−4x)Definition of gg∘g=x4−8x3+16x2−4x2+16xSimplifyg∘g=x4−8x3+12x2+16xDefinition of g
The domain of g∘g is (−∞,∞)
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