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Correct\newlineGiven \newlinef(x)f(x), find \newlineg(x)g(x) and \newlineh(x)h(x) such that \newlinef(x)=g(h(x))f(x)=g(h(x)) and neither \newlineg(x)g(x) nor \newlineh(x)h(x) is solely \newlinexx.\newlinef(x)=x+31f(x)=\sqrt{-x+3}-1\newlineAnswer\newline{:[g(x)=],[h(x)=],[]:}\{:[g(x)=\square],[h(x)=\square],[\square]:\}

Full solution

Q. Correct\newlineGiven \newlinef(x)f(x), find \newlineg(x)g(x) and \newlineh(x)h(x) such that \newlinef(x)=g(h(x))f(x)=g(h(x)) and neither \newlineg(x)g(x) nor \newlineh(x)h(x) is solely \newlinexx.\newlinef(x)=x+31f(x)=\sqrt{-x+3}-1\newlineAnswer\newline{:[g(x)=],[h(x)=],[]:}\{:[g(x)=\square],[h(x)=\square],[\square]:\}
  1. Identify Functions: Let's start by identifying the outer function g(x)g(x) and the inner function h(x)h(x) in the composition f(x)=g(h(x))f(x) = g(h(x)). We want to express f(x)f(x) as a composition of two functions, where f(x)=x+31f(x) = \sqrt{-x+3}-1. We can consider the square root and the subtraction as separate operations.
  2. Define g(x)g(x): We can choose g(x)g(x) to be the function that performs the final operation in f(x)f(x), which is subtracting 11. Therefore, we can define g(x)g(x) as g(x)=x1g(x) = \sqrt{x} - 1.
  3. Find h(x)h(x): Now we need to find h(x)h(x) such that when we apply gg to h(x)h(x), we get f(x)f(x). Since g(x)=x1g(x) = \sqrt{x} - 1, we need h(x)h(x) to be the part inside the square root of f(x)f(x). Therefore, we can define h(x)h(x) as h(x)=x+3h(x) = -x + 3.
  4. Verify Composition: Let's verify our composition by calculating g(h(x))g(h(x)) and checking if it equals f(x)f(x). We have g(x)=x1g(x) = \sqrt{x} - 1 and h(x)=x+3h(x) = -x + 3. So, g(h(x))=g(x+3)=x+31g(h(x)) = g(-x + 3) = \sqrt{-x + 3} - 1, which is indeed equal to f(x)f(x).

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