Question

Find the reciprocal; operate on numbers or on functions.

Short answer

The reciprocal operator of any function which operates on the function $f\left(x\right)$by inverting it does not represent a linear operator.

Step-by-step solution

Definition of the linear operator

An operator is said to be a linear operator if the following relations are satisfied if

$\mathit{O}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}\mathbf{=}\mathit{O}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{+}\mathit{O}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{O}\mathbf{\left(}\mathit{k}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{O}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{}$

whereis a number, andand,are numbers, functions, vectors, etc.

Operation on reciprocal function for  O(A+B)=O(A)+O(B) 

Find the Reciprocal function which operates on the function $f\left(x\right).$_.

$Q\left(x\right)=\frac{1}{f\left(x\right)}$

Evaluate role="math" localid="1658990953893" $r\left(f\right(x)+g(x\left)\right)$.

$$\begin{gathered} r\left(f\right(x)+g(x\left)\right)=\frac{1}{\left(f\right(x)+g(x\left)\right)} \\ \ne \frac{1}{f\left(x\right)}+\frac{1}{g\left(x\right)} \\ \ne r\left(f\left(x\right)\right)+r\left(f\left(x\right)\right) \end{gathered}$$

Operation on square function for O(kA)=kO(A)

Evaluate$r\left(kf\right(x\left)\right).$ .

$$\begin{gathered} r\left(kf\right(x\left)\right)=\frac{1}{kf\left(x\right)} \\ =\frac{1}{k}\frac{1}{f\left(x\right)} \\ =\frac{1}{k}r\left(f\left(x\right)\right) \\ ={}^{1}k\left(f\left(x\right)\right) \end{gathered}$$

Therefore, it is shown that the reciprocal operator of any function which operates on the function $f\left(x\right)$by inverting it does not represent a linear operator.