If both $f$ and $g$ are one-to-one functions, and $( f \circ g ) ( x ) = x ^ { 3 }$, then which of the following is/are true?
A) $( g \circ f ) ( x ) = x ^ { - 3 }$
B) $g$ is the inverse function of $f$
C) $f$ is the inverse function of $g$
D) None of the above is true.

Question

Quizplus · Accepted Answer

Given that $f$ and $g$ are one-to-one functions and $(f \circ g)(x) = x^3$, it does not necessarily imply any of the provided options. A) assumes a specific relationship between $f$ and $g$ that is not supported by the information given. B) and C) assume inverse relationships that are not directly implied by the composition being $x^3$. The fact that the composition of $f$ and $g$ results in $x^3$ does not provide enough information to determine the individual forms of $f$ and $g$, nor does it imply that one is the inverse of the other.

If Both fff And ggg Are One-To-One Functions, And (f∘g)(x)=x3( f \circ g ) ( x ) = x ^ { 3 }(f∘g)(x)=x3

If Both $f$ And $g$ Are One-To-One Functions, And $( f \circ g ) ( x ) = x ^ { 3 }$