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