Finding The Inverse Of A Rational Function (a special case)

Are you struggling to find the inverse of a rational function? Watch this video to learn how to solve this common math problem with a step-by-step example. We'll start by defining what a rational function is and then show you how to find its inverse using a clear and easy-to-understand example. You'll be surprised at how simple it can be once you learn the technique. Whether you're a math student or a professional, this video is a must-watch for anyone looking to improve their problem-solving skills. So join us on this journey to mastering inverse rational functions and take your math abilities to the next level!
#rationalfunctions #inversefunctions #mathematics #precalculus

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"Just Algebra" (by blackpenredpen) is dedicated to helping middle school, high school, and community college students who need to learn algebra. Topics include how to solve various equations (linear equations, quadratic equations, square root equations, rational equations, exponential equations, logarithmic equations, and more), factoring techniques, word problems, functions, graphs, Pythagorean Theorem, and more. We will also cover standardized test problems such as the SAT. Feel free to leave your questions in the comment!
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#justalgebra

[russelllomando]

WOW!!! I'm 71 & watch these for fun & knowledge. Now I have a headache. Amazing!

[louiezioulas]

Involution. Wow! f(f(x)) = x. Never thought of that before except for maybe y=x.

[epsi]

Interesting...
Assuming f(x) = y:

y = (-x+3)/(2x+1)
D: (-∞, -½)U(-½, +∞)
R: (need inverse to know)
2xy + y = -x+3
2xy + y + x = 3
(2y+1)x + y = 3 -- (2y+1)x = 2xy+x
(2y+1)x = -y + 3
x = (-y+3)/(2y+1) = f(y)
Since f(y) = f(f(x)) = x, f is its own inverse, meaning the domains and ranges of both f and its inverse are identical:
D: (-∞, -½)U(-½, +∞)
R: (-∞, -½)U(-½, +∞)

To verify we didn't make a mistake, we must ensure f(f(x))=x is true.
Otherwise, f is NOT its own inverse, and we must try to determine what we did wrong:

x = f(f(x))
x = (-f(x)+3) / (2f(x)+1)
x = (-(-x+3)/(2x+1)+3) / (2(-x+3)/(2x+1)+1)
x = ((x-3)/(2x+1)+(6x+3)/(2x+1)) / ((6-2x)/(2x+1) + (2x+1)/(2x+1))
x = (7x/(2x+1)) / (7/(2x+1))
x = 7x/7 -- (a/b) / (c/b) = (a/b)(b/c) = a/c
x = x

This means we didn't make a mistake when finding the inverse of f, so f is indeed its own inverse.

[JayTemple]

I worked it out: A rational function (ax + b) / (cx + d) will have (-dx + b) / (cx - a) as its inverse. Note that the "b" and "c" are in the same spot. If "a" and "d" are opposite, such as the 1 and -1 in the original example, then the -d is a and the -a is d, so the function is its own inverse.

[fasebingterfe]

Hold up, the function is its own inverse? Damn!

[TerungwaSamuel-oyiv]

inverse of a function undo the function, but according to your solving both the function and its inverse are still the same.

[LeftGuard]

I can tell you why it's correct without engaging in hyperbole.