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Math  /  Algebra

QuestionIn Exercises 15−2215-22, find f(g(x))f(g(x)) and g(f(x))g(f(x)). State the domain of each.
15. f(x)=3x+2;g(x)=x−1f(x)=3 x+2 ; g(x)=x-1
16. f(x)=x2−1;g(x)=1x−1f(x)=x^{2}-1 ; g(x)=\frac{1}{x-1}
17. f(x)=x2−2;g(x)=x+1f(x)=x^{2}-2 ; g(x)=\sqrt{x+1}
18. f(x)=1x−1;g(x)=xf(x)=\frac{1}{x-1} ; g(x)=\sqrt{x}
19. f(x)=x2;g(x)=1−x2f(x)=x^{2} ; g(x)=\sqrt{1-x^{2}}
20. f(x)=x3;g(x)=1−x33f(x)=x^{3} ; g(x)=\sqrt[3]{1-x^{3}}
21. f(x)=12x;g(x)=13xf(x)=\frac{1}{2 x} ; g(x)=\frac{1}{3 x}
22. f(x)=1x+1;g(x)=1x−1f(x)=\frac{1}{x+1} ; g(x)=\frac{1}{x-1}

Studdy Solution
Determine the domain of g(f(x))=x+1−x g(f(x)) = \frac{x+1}{-x} :
The expression x+1−x\frac{x+1}{-x} is undefined when x=0x = 0 and when x=−1x = -1 (since f(x)=1x+1 f(x) = \frac{1}{x+1} is undefined at x=−1 x = -1 ). Therefore, the domain of g(f(x)) g(f(x)) is all real numbers except x=0 x = 0 and x=−1 x = -1 .

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