Let $f\left( x \right) = {e^x} + \sin x$ be defined on the interval $x \in \left[ { - 4,0} \right]$, the odd extension of f(x) in the interval
[-4, 4]
$
A.{\text{ }}{e^{ - x}} + \sin x,x \in \left( {0,4} \right) \\
B.{\text{ }} - {e^{ - x}} + \sin x,x \in \left( {0,4} \right) \\
C.{\text{ }} - {e^{ - x}} - \sin x,x \in \left( {0,4} \right) \\
D.{\text{ }} - {e^{ - x}} - \cos x,x \in \left( {0,4} \right) \\
$
Answer
361.8k+ views
Hint: In this question we have been given a function f(x) which is defined in a certain interval and we have to find the odd extension of f(x) in the interval [-4, 4]. Odd extension means that the function breaks into a piecewise function which is defined over a specific interval, so simply find the breaking point of the given f(x) in the interval in which the odd extension is to be taken out.
Complete step-by-step answer:
Given function
$f\left( x \right) = {e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right]$
Now we have to find out the odd extension of f(x) in the interval [-4, 4]
According to odd extension property the function break into piecewise function which is defined as in the interval [-a, a]
The odd extension of f(x) is the function
\[{f_o}\left( x \right) = \left\{
f\left( x \right),{\text{ }}x \in \left[ { - a,0} \right] \\
- f\left( { - x} \right),{\text{ }}x \in \left[ {0,a} \right] \\
\right.\] So, use this property to calculate the odd extension of the given function in the interval [-4, 4]
$f\left( x \right) = {e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right]$
Now replace x with (-x) we have,
$f\left( { - x} \right) = {e^{ - x}} + \sin \left( { - x} \right)$
Now as we know $\sin \left( { - \theta } \right) = - \sin \theta $ so, use this property in the above equation we have,
$f\left( { - x} \right) = {e^{ - x}} - \sin x$
Now multiply by (-) in above equation we have,
$ - f\left( { - x} \right) = - \left( {{e^{ - x}} - \sin x} \right) = - {e^{ - x}} + \sin x$
$ - f\left( { - x} \right) = - {e^{ - x}} + \sin x,{\text{ }}x \in \left[ {0,4} \right]$
Hence the odd extension of the given function in the interval [-4, 4] is
\[{f_o}\left( x \right) = \left\{
{e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right] \\
- {e^{ - x}} + \sin x,{\text{ }}x \in \left[ {0,4} \right] \\
\right.\]
Hence option (b) is correct.
Note – Whenever we face such type of problems the key concept is to have the basic understanding of the odd extension defined over a period of interval, make sure that the interval given is only a subset of the domain of the given function otherwise there may arise a case even that the function is not defined. These concepts will help you get on the right track to get the answer.
Complete step-by-step answer:
Given function
$f\left( x \right) = {e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right]$
Now we have to find out the odd extension of f(x) in the interval [-4, 4]
According to odd extension property the function break into piecewise function which is defined as in the interval [-a, a]
The odd extension of f(x) is the function
\[{f_o}\left( x \right) = \left\{
f\left( x \right),{\text{ }}x \in \left[ { - a,0} \right] \\
- f\left( { - x} \right),{\text{ }}x \in \left[ {0,a} \right] \\
\right.\] So, use this property to calculate the odd extension of the given function in the interval [-4, 4]
$f\left( x \right) = {e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right]$
Now replace x with (-x) we have,
$f\left( { - x} \right) = {e^{ - x}} + \sin \left( { - x} \right)$
Now as we know $\sin \left( { - \theta } \right) = - \sin \theta $ so, use this property in the above equation we have,
$f\left( { - x} \right) = {e^{ - x}} - \sin x$
Now multiply by (-) in above equation we have,
$ - f\left( { - x} \right) = - \left( {{e^{ - x}} - \sin x} \right) = - {e^{ - x}} + \sin x$
$ - f\left( { - x} \right) = - {e^{ - x}} + \sin x,{\text{ }}x \in \left[ {0,4} \right]$
Hence the odd extension of the given function in the interval [-4, 4] is
\[{f_o}\left( x \right) = \left\{
{e^x} + \sin x,{\text{ }}x \in \left[ { - 4,0} \right] \\
- {e^{ - x}} + \sin x,{\text{ }}x \in \left[ {0,4} \right] \\
\right.\]
Hence option (b) is correct.
Note – Whenever we face such type of problems the key concept is to have the basic understanding of the odd extension defined over a period of interval, make sure that the interval given is only a subset of the domain of the given function otherwise there may arise a case even that the function is not defined. These concepts will help you get on the right track to get the answer.
Last updated date: 30th Sep 2023
•
Total views: 361.8k
•
Views today: 3.61k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE
