Help!!!

[jeca86]

Show that the function f(x)=ln(x+ (square root of (x^2+1))) is an odd function.

 

[Corsair_cavalier]

Definition of even function.... f(-x)=f(x)
Definition of odd function... f(-x)=-f(x)

Hopefully this helps

 

[soroban]

Hello, jeca86!

This is a real challenger . . . . I'll walk through it.

. . . . . . . . . . . . . . . . . . . . . . . . . . . _____
Show that the function f(x) = ln(x + √x^2+1) is an odd function.

We must show that: .f(-x) .= .- f(x)
. . That is, plugging in -x results in the negative of the function.
. . . . . . . . . . . . . . . . . . . . . . . . . . . _______ . . . . . . . . . .______
We plug in -x: . f(-x) .= .ln[(-x) + √(-x)² + 1] .= .ln[-x + √x^2 + 1]

. . . . . . . . . . . . . . . . . . . . . . . . . _____
Inside the log, we have: .- x + √x² + 1
. . . . . . . . . . . . . . . . . . . . . . . . . . _____
Multiply top and bottom by (x + √x² + 1):
. . . . . . . _____ . . . . . ._____
. . - x + √x² + 1 . . x + √x² + 1 . . . . -x² + (x² + 1) . . . . . . . 1. . . . . . . . . . . . . _____
. . ---------------- . --------------- . = . ------------------ . = . -------------- . = . (x + √x² + 1)^-1
. . . . . . .1 . . . . . . x + √x² + 1 . . . . . x + √x² + 1 . . . . . x + √x² + 1

. . . . . . . . . . . . . . . . . . . . . . . . . . .____ . . . . . . . . . . . . . _____
Then we have: . f(-x) . = . ln(x + √x² +1)^-1 . = . - ln(x + √x² + 1) . = . - f(x)

. . Therefore, it <u>is</u> an odd function.

 

[jeca86]

Thanx for the help. I just didn't know what to do with the natural log during the multiplication step. So do you just put the negative sign in front of the natural log automatically at the end?