Is the Function f(x) = x^5 - x^3 + 2x Invertible?

[alexs2jennisha]

Homework Statement

1. Show that the function f(x) = x⁵ -x³ +2x is invertible. Compute the derivative of f^-1
at 2.

The Attempt at a Solution

To find f^-1 I switched x and y which gave me x = y⁵ - y³ + 2y

this is where i got stuck because I am not sure how to solve for y after that step.

Can the y⁵ - y³ become y²?

Thanks

 

[LCKurtz]

Homework Statement

1. Show that the function f(x) = x⁵ -x³ +2x is invertible. Compute the derivative of f^-1
at 2.

The Attempt at a Solution

To find f^-1 I switched x and y which gave me x = y⁵ - y³ + 2y

this is where i got stuck because I am not sure how to solve for y after that step.

Lucky for you, you weren't asked to calculate $f^{-1}$, just show it exists. Can you show, for example, that $f$ is increasing?

Can the y⁵ - y³ become y²?

Of course not.

 

[member 428835]

Can the y⁵ - y³ become y²?

one really good way to see if numeric algebra works is to take a simple case and see what happens. what if $y=1$?? then you know the answer to be zero yet if you adopt your proposition we see the answer is 1, which is of course incorrect.

hope this helps for the future

 

[SammyS]

What is necessary for a function to be invertible ?