How to Find the Inverse of a Polynomial: Step-by-Step Guide

The point (1, f(1)) is on the graph of f(x)= 2x^3+ 5x+ 3. So when x= 1, y= 2(1)^3+ 5(1)+ 3= 2+ 5+ 3= 10. That point is (1, 10). Now, from x= f^{-1}(y), 1= f^{-1}(10). So f^{-1}(10)= 1.
  • #1
HopelessCalc
1
0

Homework Statement



Let f(x) = 2x^3 + 5x + 3
Find the inverse at f^-1(x) = 1

Homework Equations


N/A

The Attempt at a Solution



The only way that I know how to solve inverses is by solving for X, then replacing it by Y. Then I supposed I would sub 1 into the inverted polynomial. However I'm not sure how to solve for X.
My attempt:
y - 3 = 2x^3 + 5x
(y - 3)/x = x(2x^2 + 5)

Then, hopelessness. Any info will be extremely helpful for my test this Friday. Thank you.
 
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  • #2
Your statement is a bit unclear -- I'm interpreting you to say that you want to find x = f-1(1) , that is, what value of x gives f(x) = 1 ?

I don't think they want you to solve the cubic equation [itex]2x^{3} + 5x + 3 = 1 [/itex] (unless they taught you how to do so in your course). This doesn't come out nicely at all...

I'm wondering if they're asking that if you had the function y = f-1(x)*, what value of x would give y = 1 ? This is equivalent to asking what f(1) equals: the value of f(1) is the number that f-1(x) would "take back" to 1 .*whatever that is exactly -- it wouldn't be pretty; instead, we will work with what is called the "implicit function"
 
  • #3
buddy think again;
the answer is f(1)...
 
  • #4
Yes, as both dynamicsolo and abhishek ghos are saying, you are completely misunderstanding the problem. The problem does NOT as you to find the inverse function, which would be extremely complex. It only asks you to find the single value [itex]f^{-1}(1)[/itex].

Use the fact that if [itex]f(x)= y[/itex] then [itex]x= f^{-1}(y)[/itex].
 

FAQ: How to Find the Inverse of a Polynomial: Step-by-Step Guide

What is the inverse of a polynomial?

The inverse of a polynomial is a new polynomial that, when multiplied by the original polynomial, produces a constant term of 1. In other words, the inverse of a polynomial is the "opposite" of the original polynomial.

How do you find the inverse of a polynomial?

To find the inverse of a polynomial, you must first factor the polynomial completely. Then, you can use the factored form to determine the inverse by switching the positions of the factors and changing the signs of all the coefficients.

Why is finding the inverse of a polynomial important?

Finding the inverse of a polynomial is important because it allows us to solve equations involving polynomials more easily. It also allows us to find the roots of a polynomial and understand its behavior more clearly.

Can every polynomial have an inverse?

No, not every polynomial has an inverse. For a polynomial to have an inverse, it must be a non-constant polynomial with a nonzero constant term. Additionally, the polynomial must be one-to-one, meaning that every element in the range has a unique corresponding element in the domain.

Can the inverse of a polynomial be a polynomial of a different degree?

Yes, the inverse of a polynomial can be a polynomial of a different degree. In fact, most of the time, the inverse of a polynomial will have a different degree than the original polynomial. This is because the inverse polynomial is obtained by switching the positions of the factors and changing the signs of the coefficients, which can result in a different degree polynomial.

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