Q9P Question: Find the square; opera... [FREE SOLUTION]

Chapter 3: Q9P (page 130)

Question: Find the square; operate on numbers or on functions.

Short Answer

Expert verified

The square of any function which operates on the function $f (x)$ by squaring 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 $O (A + B) = O (A) + O (B) a n d O (k A) = k O (A) .$ .

Where, K is a number, and A and, B are numbers, functions, vectors, etc.

Operation on square function for O(A+B)=O(A)+O(B) :

The square function which operates on the function is given by

$Q (x) = [f (x)]^{2}$

It is known that

${(f (x) + g (x))}^{2} = f^{2} (x) + 2 f (x) g (x) + g^{2} (x) = {[f (x)]}^{2} + {[g (x)]}^{2} + 2 f (x) g (x) + g^{2} (x) = {[f (x)]}^{2} + {[g (x)]}^{2}$

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

Evaluate the square of $(k f (x))^{2}$ .

$(k f (x))^{2} = k^{2} f^{2} (x) = k^{2} {[f (x)]}^{2} 鈮� k^{2} {[f (x)]}^{2}$

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Question: Find the square; operate on numbers or on functions.

Short Answer

Step by step solution

Definition of the linear operator:

Operation on square function for O(A+B)=O(A)+O(B) :

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Linear Momentum

Kinematics Physics

Electrostatics

Geometrical and Physical Optics

Dynamics

Space Physics

Study anywhere. Anytime. Across all devices.