Find Inverse of Rational Function

In summary, to find the inverse of a function, you can replace x with y and solve for y. The inverse of a function is found by swapping the x and y coordinates of a point and reflecting it across the line y=x. In order for a function to have an inverse, it must be one-to-one, meaning that for every x there is exactly one y and vice versa. If the function is too complicated to find its inverse by hand, a graphing calculator or computer system can be used. However, it is important to note that not all functions have inverses, and it is crucial to understand what it means for a function to have an inverse.
  • #1
mathdad
1,283
1
Find the inverse of
f(x) = 2/(x - 3).

Let y = f(x)

y = 2/(x - 3)

Replace y for x.

x = 2/(y - 3)

x(y - 3) = 2

Solve for y.

xy - 3x = 2

xy = 2 + 3x

y = (2 + 3x)/x

Replace y with f^-1 (x).

f^-1(x) = (2 + 3x)/x

1. Is f^-1(x) the inverse of f(x)?

2. What does f(x) and f^-1(x) look like together on the same xy-plane?
 
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  • #2
Re: Find Inverse of Rational Number

RTCNTC said:
1. Is f^-1(x) the inverse of f(x)?

If you did the work correctly, then you should find:

\(\displaystyle f\left(f^{-1}(x)\right)=f^{-1}\left(f(x)\right)=x\)

RTCNTC said:
2. What does f(x) and f^-1(x) look like together on the same xy-plane?

A function and it inverse will appear to be reflected across the line $y=x$.
 
  • #3
About the reflection part, if we have some point on a function $(x,y)$, then the point on the inverse corresponding to this point is naturally $(y,x)$. The mid-point of these points is:

\(\displaystyle \left(\frac{x+y}{2},\frac{x+y}{2}\right)\)

This point is of course on the line $y=x$.

And the slope of the line segment between them is:

\(\displaystyle m=\frac{x-y}{y-x}=-1\)

The two points are the same perpendicular distance from the line $y=x$.

Thus, we see that a point reflected across the line $y=x$ will simply require swapping $xy$ coordinates, which is what the inverse is.
 
  • #4
So, the inverse of (x, y) is simply (y, x). What if the function is too complicated to find its inverse by hand? Must we then rely on a computer system or graphing calculator?
 
  • #5
RTCNTC said:
So, the inverse of (x, y) is simply (y, x). What if the function is too complicated to find its inverse by hand? Must we then rely on a computer system or graphing calculator?
If the function has an inverse that is also a function, then there can only be one y for every x. In two dimensions. Higher dimensions, yes I would use numerical compilers.

A one-to-one function, is a function in which for every x there is exactly one y and for every y, there is exactly one x.

A one-to-one function has an inverse that is also a function.

There are functions which have inverses that are not functions. There are also inverses for relations. For the most part, we disregard these, and deal only with functions whose inverses are also functions.

Later on in abstract algebra this is one of the key ingredients for an isomorphism. Its the basis for a lot of topology classes, analysis classes, combinatorics. Its really a fundamental necessity to understand what it means for a function to be an inverse. I.E. the identitiy is one key.
 
  • #6
DrWahoo said:
If the function has an inverse that is also a function, then there can only be one y for every x. In two dimensions. Higher dimensions, yes I would use numerical compilers.

A one-to-one function, is a function in which for every x there is exactly one y and for every y, there is exactly one x.

A one-to-one function has an inverse that is also a function.

There are functions which have inverses that are not functions. There are also inverses for relations. For the most part, we disregard these, and deal only with functions whose inverses are also functions.

Later on in abstract algebra this is one of the key ingredients for an isomorphism. Its the basis for a lot of topology classes, analysis classes, combinatorics. Its really a fundamental necessity to understand what it means for a function to be an inverse. I.E. the identitiy is one key.

1. Thank you for the data.

2. I am doing a self-study of math.

3. I am not a student.

4. I am a middle-aged man.

5. I do not need to go beyond Calculus 3, which is my ultimate goal that will be accomplished in time. Linear algebra and beyond is not my concern. Abstract algebra is so not possible given my age and interest.
 
  • #7
RTCNTC said:
So, the inverse of (x, y) is simply (y, x). What if the function is too complicated to find its inverse by hand? Must we then rely on a computer system or graphing calculator?

Not exactly. The function y could possibly not have an inverse at all. If you are in two dimensions and the function is one- to- one then yes, your logic holds.

The typical vertical line test tells you by graphing a function is in fact a function (vertical line test). Use the horizontal line test to check if the graph of the inverse is indeed true. This follows from the result of the transformation MikeFL discussed about the reflection.

- - - Updated - - -

RTCNTC said:
1. Thank you for the data.

2. I am doing a self-study of math.

3. I am not a student.

4. I am a middle-aged man.

5. I do not need to go beyond Calculus 3, which is my ultimate goal that will be accomplished in time. Linear algebra and beyond is not my concern. Abstract algebra is so not possible given my age and interest.

Very good questions! It is always good to keep old topics fresh. I am a professor and I tend to forgot little details at times. Just remember practice, practice, practice. Good luck to you in those courses!
 
  • #8
DrWahoo said:
Not exactly. The function y could possibly not have an inverse at all. If you are in two dimensions and the function is one- to- one then yes, your logic holds.

The typical vertical line test tells you by graphing a function is in fact a function (vertical line test). Use the horizontal line test to check if the graph of the inverse is indeed true. This follows from the result of the transformation MikeFL discussed about the reflection.

- - - Updated - - -
Very good questions! It is always good to keep old topics fresh. I am a professor and I tend to forgot little details at times. Just remember practice, practice, practice. Good luck to you in those courses!

Thanks. Not courses but self-study.
 

FAQ: Find Inverse of Rational Function

What is the definition of a rational function?

A rational function is a function that can be expressed as the quotient of two polynomial functions, with the denominator not equal to zero. It is also known as a ratio of two polynomials.

Why is it important to find the inverse of a rational function?

Finding the inverse of a rational function allows us to solve equations involving the function, as well as find the domain and range of the function. It also helps in graphing the function and understanding its behavior.

How do you find the inverse of a rational function?

To find the inverse of a rational function, switch the positions of the numerator and denominator, and then solve for the new numerator. The resulting function will be the inverse of the original rational function.

Can all rational functions have an inverse?

No, not all rational functions have an inverse. For a rational function to have an inverse, it must pass the horizontal line test, which means that no horizontal line should intersect the graph of the function more than once.

What is the difference between finding the inverse of a rational function and finding the reciprocal of a rational function?

Finding the inverse of a rational function involves switching the positions of the numerator and denominator and solving for the new numerator. Finding the reciprocal, on the other hand, involves simply flipping the fraction. The reciprocal of a rational function is not necessarily its inverse, unless the function is a one-to-one function.

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