danika concludes that the following functions are inverses of each other because ( f(g(x)) = x ). do you…
danika concludes that the following functions are inverses of each other because ( f(g(x)) = x ). do you agree with danika? explain your reasoning.\n( f(x) = |x| )\n( g(x) = -x )
Answer
Explanation:
Step1: Calculate (g(f(x)))
To check if two functions (f(x)) and (g(x)) are inverses, we need to verify both (f(g(x)) = x) and (g(f(x))=x). We know (f(x)=\vert x\vert) and (g(x)=-x). First, find (g(f(x))): Substitute (f(x)) into (g(x)), so (g(f(x))=g(\vert x\vert)) Since (g(t)=-t) (where (t = \vert x\vert)), then (g(\vert x\vert)=-\vert x\vert)
Step2: Analyze the result
For (x>0), (\vert x\vert=x), and (g(f(x))=-x\neq x). For (x < 0), (\vert x\vert=-x), and (g(f(x))=-(-x)=x). But for a general (x\in R), (g(f(x))=-\vert x\vert\neq x) (when (x>0)).
Answer:
No, Danika is incorrect. While (f(g(x))=\vert -x\vert=\vert x\vert). For (x>0), (f(g(x)) = x), but to be inverses, (g(f(x))) must also equal (x) for all (x) in the domain. Since (g(f(x))=-\vert x\vert\neq x) when (x>0), the functions (f(x)) and (g(x)) are not inverses of each other.