Suppose that f(x)=√x and g(x)=2x−4 find f∘g, g∘f, f∘f and g∘g and their domains
f∘g=f(g(x))=f(2x−4)=√2x−4g∘f=g(f(x))=g(√x)=2√x−4f∘f=f(f(x))=f(√x)=√√x=4√xg∘g=g(g(x))=g(2x−4)=22x−4−4=2(x−4)2−4(x−4)=2(x−4)2−4x+16=2(x−4)−4x+18=2(x−4)−2(2x−9)=−(x−4)2x−9
To find the domain of f∘g, we want...
x−4>0Add 4x>4
Thus, the domain of f∘g is (4,∞)
To find the domain of g∘f, we want √x−4≠0. The denominator is zero when...
√x−4=0Add 4√x=4Square both sidesx=16
Also, the function involves square root that is defined for only positive values. Thus, the domain of g∘f is [0,16)⋃(16,∞)
To find the domain of f∘f, recall that the function with even roots is defined only for positive values of x. Thus, the domain of f∘f is [0,∞)
To find the domain of g∘g, we want 2x−9≠0 the denominator is zero when...
2x−9=0Add 92x=9Divide 2x=92
Thus, the domain of g∘g is (−∞,92)⋃(92,∞)
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