Question

(a) Determine the value of \({f^{ - 1}}(17)\) if\(f\) is one-to-one and \(f(6) = 17\).

(b) Determine the value of \(f(2)\) if\(f\) is one-to-one and \({f^{ - 1}}(3) = 2\).

Short answer

(a) The resultant answer is\({\kern 1pt} {\kern 1pt} {f^{ - 1}}(17) = 6\).

(b) The resultant answer is \({\kern 1pt} {\kern 1pt} f(2) = 3\).

Step-by-step solution

Given data

The given function is\({\kern 1pt} {\kern 1pt} f(6) = 17\) and \({\kern 1pt} {\kern 1pt} {f^{ - 1}}(3) = 2\).

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Simplify the expression

(a)

It is given that the function is one-to-one and \({\kern 1pt} {\kern 1pt} f(6) = 17\).

According to the definition of an inverse function, \({f^{ - 1}}(y) = x\) if and only if \(f(x) = y\) where \(f\) is one-to-one.

The value of \({f^{ - 1}}(17) = 6\) since \({\kern 1pt} {\kern 1pt} f(6) = 17\).

Thus, it can be concluded that the value of \({\kern 1pt} {\kern 1pt} f(6) = 17\).

Simplify the expression

(b)

It is given that the function is one-to-one and \({f^{ - 1}}(3) = 2\).

According to the definition of an inverse function, \({f^{ - 1}}(y) = x\) if and only if \(f(x) = y\) where \(f\) is one-to-one.

The value of \(f(2) = 3\) since \({f^{ - 1}}(3) = 2\).

Thus, it can be concluded that the value of \(f(2) = 3\).