Which pairs of functions are inverses? (a) f(x)=-3 x+2 ; g(x)=(2)/(3)-(1)/(3) x (b) F(x)=2 x+1 ;

Which pairs of functions are inverses? (a) f(x)=-3 x+2 ;...

Key Concepts

Inverse Functions

Inverse functions are pairs of functions where one function undoes the effect of the other. In mathematical terms, when you compose a function with its inverse, the result is the identity function, meaning that the input remains unchanged. This concept is fundamental in solving equations and understanding how functions can be reversed.

Function Composition

Function composition involves applying one function to the result of another. When checking for inverses, one typically verifies that the composition of the two functions in either order yields the identity function. This property is crucial in determining whether two functions are true inverses of each other.

One-to-One Functions

A one-to-one (injective) function is one where each element of the function’s domain maps to a unique element in the codomain. This property is necessary for a function to have an inverse because it ensures that the inverse will also be a function, meaning every output corresponds to exactly one input.

Linear and Polynomial Function Inversion

For linear functions, finding an inverse generally involves solving a simple equation to express the input in terms of the output, which is feasible when the function is one-to-one. Similarly, certain polynomial functions, like cubic functions and their cube roots, because of their monotonicity, allow for inversions, as the operations they represent can undo one another.