Answer

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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.

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