SOME SIMPLE F^-1 functions cant solve

[kadmany]

SOME SIMPLE F^-1 functions! can't solve! :(

Homework Statement

Three functions :

1- f(x)=e^x


2- f(x)=sin(e^x)

3- f(x)=e^sin(x)

Homework Equations

I must find each functions [f^-1]

The Attempt at a Solution

1- I found that f^-1(y)=In(y)

2- TOTALLY NO IDEA HOW!

3- y=e^sin(x) /IN ==> In(y)=sin(x),ok then? if I use arc sin then how I use it ?

Thanks!

 

[HallsofIvy]

Homework Statement

Three functions :

1- f(x)=e^x


2- f(x)=sin(e^x)

3- f(x)=e^sin(x)

Homework Equations

I must find each functions [f^-1]

The Attempt at a Solution

1- I found that f^-1(y)=In(y)

That's NOT an "I", it is an "l" for "logarithm". Yes, one way of defining "ln(x)" is as the inverse to the $e^x$.

2- TOTALLY NO IDEA HOW!

The same way you approached (3). The inverse of sine is arc sin so if $y= sin(e^x)$ then $arcsin(y)= e^x$ and then use the logarithm to get rid of the exponential: ln(arcsin(y))= x.

3- y=e^sin(x) /IN ==> In(y)=sin(x),ok then? if I use arc sin then how I use it ?

Again, it is "ln", not "In". Yes, if $y= e^{sin(x)}$ then $ln(y)= sin(x)$. And now you apply "arcsin" to both sides: $arcsin(ln(y))= x$

Thanks!

 

[kadmany]

That's NOT an "I", it is an "l" for "logarithm". Yes, one way of defining "ln(x)" is as the inverse to the $e^x$.


The same way you approached (3). The inverse of sine is arc sin so if $y= sin(e^x)$ then $arcsin(y)= e^x$ and then use the logarithm to get rid of the exponential: ln(arcsin(y))= x.


Again, it is "ln", not "In". Yes, if $y= e^{sin(x)}$ then $ln(y)= sin(x)$. And now you apply "arcsin" to both sides: $arcsin(ln(y))= x$

Oh real thanks man! now I understand how to use arcsin !