Chapter 3: Q10P (page 130)

Find the reciprocal; operate on numbers or on functions.

Short Answer

Expert verified

The reciprocal operator of any function which operates on the function $f (x)$ 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

$O (A + B) = O (A) + O (B) a n d O (k A) = k O (A)$

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 $f (x) .$ _.

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

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

$r (f (x) + g (x)) = \frac{1}{(f (x) + g (x))} \neq \frac{1}{f (x)} + \frac{1}{g (x)} \neq r (f (x)) + r (f (x))$

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

Evaluate $r (k f (x)) .$ .

$r (k f (x)) = \frac{1}{k f (x)} = \frac{1}{k} \frac{1}{f (x)} = \frac{1}{k} r (f (x)) = {}^{1}k (f (x))$

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

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Find the reciprocal; operate on numbers or on functions.

Short Answer

Step by step solution

Definition of the linear operator

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

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

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