Question

Assertion :The function $\sinh \left(x\right)$ is an odd function. Reason: The function $\sinh \left(x\right)$ is symmetric with respect to the origin, and $\sinh \left(0\right)=0$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion

. $\because \sinh x$ is an odd function

$f\left(x\right)=\sinh x=\frac{e^x-e^{-x}}{2}$

$f(-x)=\sinh (-x)=\frac{e^{-x}-e^{-(-x)}}{2}=\frac{-(e^x-e^{-x})}{2}=-f\left(x\right)$

$\because f(-x)=-f\left(x\right)$ , which shows $f\left(x\right)$ is an odd function.

Reason

Yes $\sinh x$ is symmetrical w.r.t. $0$

And it is also the condition to be an odd function.

And odd function is symmetric at origin.

$\cosh e^x$ is even function

$f\left(x\right)=\cosh x=\frac{e^x+e^{-x}}{2}$

$f(-x)=\cosh (-x)=\frac{e^{-x}+e^{-(-x)}}{2}=\frac{-(e^x+e^{-x})}{2}=f\left(x\right)$

Hence function is even function.

The graph of $\cosh x$ s which is symmetric about $y$ axis.

And the condition of even function is $f(-x)=-f\left(x\right)$

and symmetric about $y$ axis

Hence Assertion and Reason both are correct and Reason is correct explanation of Assertion.