Integral and inverse function: f(x) = e^x + x - 1

[mikeyWay]

Given \(\displaystyle f:\, \mathbb{R}\, \rightarrow\, \mathbb{R},\, f(x)\, =\, e^x\, +\, x\, -\, 1,\) and \(\displaystyle f^{-1}(x)\) being the inverse of \(\displaystyle f(x),\) calculate the following:

. . . . .\(\displaystyle \displaystyle \left(f^{-1}\right)'(0)\, +\, \int_0^e\, f^{-1}(x)\, dx\)

My question...

How can i find the inverse of this function?


First post. Thanks for help.

 

[Steven G]

Given \(\displaystyle f:\, \mathbb{R}\, \rightarrow\, \mathbb{R},\, f(x)\, =\, e^x\, +\, x\, -\, 1,\) and \(\displaystyle f^{-1}(x)\) being the inverse of \(\displaystyle f(x),\) calculate the following:

. . . . .\(\displaystyle \displaystyle \left(f^{-1}\right)'(0)\, +\, \int_0^e\, f^{-1}(x)\, dx\)

My question...

How can i find the inverse of this function?

First post. Thanks for help.

Can you find the same area you are asked to find for the inverse using f(x)? Hint: the answer is yes.

So figure out what the limits of integration would be for f(x) and integrate.

 

[mikeyWay]

Can you find the same area you are asked to find for the inverse using f(x)? Hint: the answer is yes.

So figure out what the limits of integration would be for f(x) and integrate.

I don't understand... Can you be more explicit?

 

[Dr.Peterson]

Given \(\displaystyle f:\, \mathbb{R}\, \rightarrow\, \mathbb{R},\, f(x)\, =\, e^x\, +\, x\, -\, 1,\) and \(\displaystyle f^{-1}(x)\) being the inverse of \(\displaystyle f(x),\) calculate the following:

. . . . .\(\displaystyle \displaystyle \left(f^{-1}\right)'(0)\, +\, \int_0^e\, f^{-1}(x)\, dx\)

My question...

How can i find the inverse of this function?

First, to make sure I'm reading the slightly fuzzy picture, you are asked to calculate \(\displaystyle \displaystyle \left(f^{-1}\right)'(0) + \int_0^ef^{-1}(x)dx\), right?

The main thing to know is that you don't need to find the inverse function; that's good, because you can't find an algebraic expression for f'.

You are meant to use other ideas to evaluate the expression. First, you can find f'; how is (f^-1)' related to that? Second, you can sketch the region whose area is found by the integral of the inverse function; it will help to find what f(0) and f(1) are, and use that to find f^-1(0) and f^-1(e), which you will need. Then you can think about how else you can find the same area (maybe using horizontal rectangles rather than vertical rectangles as your elements).

Give it a try, and ask specific questions wherever you need to. We want to help you solve this yourself, which means we'll do as little as we can for you, and therefore need to know specifically where you need help and what you can do on your own.

 

[Steven G]

I don't understand... Can you be more explicit?

Draw any invertible function (you don't even need to know the function you are drawing!)and pick any area under that curve. Now draw the inverse of that function (it's the mirror image across the line y=x). Can you find that same area?

 

[mikeyWay]

Two days of thinking and still nothing...

I don't understand how helps me what you said... What is the link between f' and (f^-1)'....

f(0)=0
f(1)=e

and how these help me to find f^-1(0) and f^-1(e)?

"horizontal rectangles rather than vertical rectangles as your elements" - I don't think I know all about it...

the graph and inverse are something like that... i think...