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Algebraically verify whether f and g are inverses of each other or not: {:[f(x)=sqrt(-8+x)","" domain "[8","oo)],[g(x)=-8+x^(2)","" domain "[0","oo)]:}

Algebraically verify whether f f and g g are inverses of each other or not:\newlinef(x)=−8+x, domain [8,∞)g(x)=−8+x2, domain [0,∞) \begin{array}{l} f(x)=\sqrt{-8+x}, \text { domain }[8, \infty) \\ g(x)=-8+x^{2}, \text { domain }[0, \infty) \end{array}

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Full solution

Q. Algebraically verify whether f f and g g are inverses of each other or not:\newlinef(x)=−8+x, domain [8,∞)g(x)=−8+x2, domain [0,∞) \begin{array}{l} f(x)=\sqrt{-8+x}, \text { domain }[8, \infty) \\ g(x)=-8+x^{2}, \text { domain }[0, \infty) \end{array}
  1. Plug and Simplify: Step 11: Plug g(x)g(x) into f(x)f(x) and simplify to see if the result is xx.
  2. Check Domain of f(g(x))f(g(x)): Step 22: Check the domain of f(g(x))f(g(x)). The domain of gg is [0,∞)[0, \infty), so the domain of f(g(x))f(g(x)) is also [0,∞)[0, \infty).
  3. Plug and Simplify: Step 33: Plug f(x)f(x) into g(x)g(x) and simplify to see if the result is xx.
  4. Check Domain of \newlineg(f(x))g(f(x)): Step 44: Check the domain of \newlineg(f(x))g(f(x)). The domain of \newlineff is \newline[8,∞)[8, \infty), so the domain of \newlineg(f(x))g(f(x)) is also \newline[8,∞)[8, \infty).
  5. Verify Domains for Inverses: Step 55: Verify if the domains of ff and gg allow for them to be inverses.

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