The functions in Exercises 11-28 are all one-to-one. For each fun...

Question

The functions in Exercises 11-28 are all one-to-one. For each function,
a. Find an equation for f^-1(x), the inverse function.
b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x +4)/(x-2)

Accepted Answer

Hello. Today we're going to be finding the inverse of our given F. Of X function. And verifying that inverse. So the given F of X function is F. Of X. Is equal to the quantity X plus eight over x minus three. Now the first step into finding the inverse is to rewrite F of X as Y. And then the second step is going to be to switch the positions of X and Y. And doing so is going to give me X is equal to Y plus eight over y minus three. And now in order to find the inverse, we need to go ahead and solve for the variable Y. That's going to give us our inverse function. So in order to start that we're gonna multiply both sides of the equation by y minus three because we want to get rid of this denominator. Doing so is going to give me X times the quantity y minus three is equal to Y plus eight. Now the next step is to go ahead and distribute this x into the quantity wipe -3. Doing so is going to give me X Y minus three. X. Is equal to Y plus eight. Now since we want to sulfur Y let's go ahead and get all the terms that involve Y to one side of the equation and anything that doesn't have Y to the other side. So I'm gonna go ahead and add three X to both sides of the equation. And I'm going to subtract Y to both sides. Doing so is going to give me X Y minus Y. Is equal to three, X plus eight. Now, on the left hand side of this equation, notice how both of the terms have a common factor of why? So we can go ahead and factor out a Y from the left hand side of the equation. Doing so is going to give me why times the quantity X minus one is equal to three X plus eight. Now, since I want to get Y by itself, I'm going to divide both sides of this equation by the quantity X -1. And that's going to give me why is equal to three X plus eight over x minus one. And this is going to be our inverse function F inverse. And this is the inverse where X cannot equal to one. Because if you were to plug in one into the, into the inverse, you're going to get a value of zero in the denominator. So, and that's going to produce an undefined value. So we found our inverse. But now we need to verify that the inverse is the inverse. And in order to do that we need to find two composite functions F of f, inverse of X and f inverse of F of X. When we find these two composite functions, if they are equal to each other and produce the value of X, then that's going to verify that they are in verses of each other. So let's go ahead and start by finding F of f inverse of x. So that means that we're gonna go ahead and plug in our inverse function into our original function and that's going to give us three X plus eight over x minus one plus eight. All over three, X plus eight over x minus one minus three. Now we need to go ahead and simplify the numerator and denominator. So simplifying the numerator and combining the terms together is going to give us the value 11 X over x minus one. And doing the same to the denominator is going to give us 11 over x minus one in order to simplify the complex fraction. We're going to multiply by the reciprocal of the denominator to both the numerator and denominator. That reciprocal is going to be x minus 1/11. And that's going to completely cancel out this denominator. And for the numerator we can go ahead and cancel the quantities X minus one and 11. And that is going to leave us with X which verifies that F of f inverse of X is an inverse. Now we need to go ahead and find the composite function F inverse of F of X. And that's going to give us three times the quantity X plus eight over X minus three plus eight. All over X plus eight over x minus three minus one. Now again we're going to simplify the numerator and denominator and combine the terms together. Doing so is going to give us a numerator of 11, X over X -3. And all of that is going to be over 11 over X -3. Now, in order to simplify the complex fraction, I'm going to multiply by the reciprocal of the denominator to both the numerator and denominator. And that reciprocal is going to be X -3/11. That's going to completely cancel out this complex denominator. And for the numerator we can go ahead and cancel out the X -3 and the 11 and that is just going to leave us with X. So f of f inverse of X is equal to f inverse of F of X and both are equal to X, which verifies that our inverse works with that being said, R f inverse is three, X plus eight over x minus one, where X cannot equal to one. And the answer to this problem is going to be be So I hope this video helps you understand the entire process of finding an inverse and I'll go ahead and see you all in the next video

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The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x +4)/(x-2)