Calculus: Solving Inverse Functions for f(x)=2x^3+3x^2+7x+4

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SUMMARY

The discussion focuses on solving inverse functions for the cubic function f(x) = 2x^3 + 3x^2 + 7x + 4. Participants highlight that while the cubic formula can yield an inverse, it is overly complex for practical use. Instead, they recommend using the relationship f^(-1)(f(x)) = x and applying the chain rule to find the derivative of the inverse function, specifically f^(-1)'(a) at a = 4. This approach simplifies the problem by finding the corresponding x value for which f(x) equals 4, rather than deriving the general inverse.

PREREQUISITES
  • Understanding of inverse functions in calculus
  • Familiarity with the cubic formula
  • Knowledge of differentiation and the chain rule
  • Basic skills in solving polynomial equations
NEXT STEPS
  • Study the application of the chain rule in calculus
  • Learn about the properties of cubic functions and their inverses
  • Explore methods for finding roots of cubic equations
  • Research practical applications of inverse functions in real-world problems
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Students and educators in calculus, mathematicians dealing with polynomial functions, and anyone looking to deepen their understanding of inverse functions and their derivatives.

ace123
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Hi I'm trying to remember inverse functions for calculus but I'm having a few problems. So any help would be appreciated.

f(x)= 2x^3 + 3x^2 + 7x+ 4

So I have no clue how to solve this for the inverse. I know how to do basic ones. But I've forgotten these kind. So can i just get a step in the right direction.
 
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You can get an inverse expression for a general cubic only by using the cubic formula. I DON'T recommend this. It's too complicated. What's the actual problem you need to solve? You probably don't need an explicit formula.
 
I'd agree, it would depend on the problem. You could use the cubic formula, but its pretty nasty.
 
Well I was hoping their was a nice way of doing b/c I knew about the cubic formula but wasn't about to use it for this. The actual question though was to find (f^-1)'(a) and the a= 4. I thought I had to find the inverse of the f(x) to solve the problem. Is there another way?
 
Yes there is. f^(-1)(f(x))=x. Differentiate both sides and use the chain rule. f^(-1)'(f(x))*f'(x)=1. So f^(-1)'(f(x))=1/f'(x). If you want to use this at f(x)=a=4, You still have to find a value of x such that f(x)=4. But that's a lot easier problem than finding the general inverse.
 
Oh, okay I see what your saying I didn't think about using f(x)=a=4. Thank you
 

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