Inverse Function: One-to-One Property and Area Calculation [f(x)=x^5+x^3+x]

[gruffy123]

First I need to show that f(x)=x^5+x^3+x is one to one
Then I need to calculate an area [3,42]... for which I need of the inverse of that fuction.
I found all sorts of graphing functions but I can't find anything to give me values or an actual function... Please please please if anyone can tell me how to get the inverse of that orginal function I will love you!

 

[Ben Niehoff]

There's an easy way you can get the area under an inverse function without actually inverting the function. Try drawing a graph of the function, and see if you can figure it out.

Hint: If you're integrating from y1 to y2, look at the rectangle formed by (x1, y1) and (x2, y2). What can you say about the areas of each part of this rectangle, on either side of y=f(x)?

 

[HallsofIvy]

First I need to show that f(x)=x^5+x^3+x is one to one

It helps a lot to know that f'(x)= 5x⁴+ 3x^2+ 1 is always positive!

Then I need to calculate an area [3,42]... for which I need of the inverse of that fuction.
I found all sorts of graphing functions but I can't find anything to give me values or an actual function... Please please please if anyone can tell me how to get the inverse of that orginal function I will love you!

Ben Niehoff's hint is excellent. DONT' get the inverse- use "symmetry". The graph of y= f(x) is just the graph of y= f^-1(x) "reflected" through the line y= x: inverting a function "swaps" x and y.
When you say "calculate and area [3, 42]" do you mean the area under the graph of y= f^-1(x) and between x= 3 and x= 42? That would correspond to x and y in f(x) such that f(x)= 3 and f(x)= 42. Can you find those values? (Hint: when faced with a really HARD equation, search for really EASY answers!)