#### To determine

**To find:**

We have to show that the function fx=lnx+x2+1 is an odd function.

#### Answer

*f(x)* is an odd function

#### Explanation

**Calculation:**

Since, we know that the function is an odd function, if f-x=-fx.

Consider the function

f-x=ln-x+x2+1

=ln-x+x2+1×x+x2+1x+x2+1

Since,a2-b2=a+ba-b. So

f-x=lnx2+1-x2x+x2+1=ln1x+x2+1

We know that

-lnx=lnx-1=ln1x

So,

f-x=lnx+x2+1-1=-lnx+x2+1=-f(x)

Therefore *, f(x)* is an odd function.

**Conclusion:**

*f(x)* is an odd function

#### To determine

**To find:**

We have to find the inverse function of fx=lnx+x2+1.

#### Answer

The inverse function is y=sinhx.

#### Explanation

**Calculation:**

First, we write the function as

y=lnx+x2+1

Now, we solve the equation for *x*. For this, first we apply the exponential function on both sides of the equation.

ey=elnx+x2+1

Or

ey=x+x2+1

Or

x2+1=ey-x……(A)

Now, taking the square of the equation (A), we have

x2+1=e2y+x2-2eyx

Now, simplifying the equation

e2y-1=2eyx

Or

x=ey-e-y2=sinhy

Now, change *x* and *y*

*y=sinhx*

So, the inverse function is y=sinhx.

**Conclusion:**

The inverse function is y=sinhx.