In Exercises 67–72, you will explore some functions and their ...

Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Accepted Answer

Hello. In this video we are given the function f of x equal to 5 x + 1 divided by 3 x minus 4, and it is defined on the domain from -1 to 1. We want to find the inverse function. So, in order to find the inverse function, let's first write down the given function. We have F of X is equal to 5 x + 1 divided by 3 X minus 4. Now the first step in finding the inverse is to swap the symbol of F of X with Y. So we have Y is equal to 5 x + 1 divided by 3 x minus 4. Then what we need to do is we need to swap the positions of both X and Y. That is going to leave us with X is equal to 5 Y plus 1 divided by 3 Y minus 4. Now what we need to do is we need to isolate and solve for why. So what we will do first is simplify the rational function. We will start by multiplying both sides of the equation by the denominator 3y minus 4. That will leave us with x multiplied by the quantity 3 Y minus 4 equal to 5 Y minus 1. We will then distribute x into the quantity 3 Y minus 4. That will leave us with 3 x y minus 4 x is equal to 5 Y minus 1. Now, what we want to do is you want to move all the terms that involve Y to one side of the equation and all terms that don't have Y to the other side of the equation. So what we're going to do is we're going to subtract 5 Y to both sides of the equation. And we are going to add 4 x to both sides of the equation. That will leave us with 3 XY minus 5Y is equal to 4 X minus 1. Notice that both of the terms on the left hand side have a factor of Y. So what we can go ahead and do is we can factor out a Y from the left hand side of the equation. That will leave us with y multiplied by the quantity 3 x minus 5 equal to 4 x minus 1. And finally, what we are going to do is we are going to divide both sides both sides of this equation by 3 X minus 5. That will leave us with y is equal to 4 x minus 1 divided by 3 x minus 5. Now, looking back at my work, I did make a small mistake here. In this step, we did have a symbol in the numerator that tells us we have a sum, and I accidentally wrote that as a negative sign on the right-hand side. So what I need to do is I need to fix that by changing 5 Y minus 1 into 5Y + 1. So the numerator of our inverse function should have a sum. So our inverse function should be F inverse Y or F inverse X equal to 4 x + 1 divided by 3 X minus 5. This is going to be our inverse function, but now what we need to do is we need to go ahead and define the domain that the inverse function exists on. Now, in order to find the domain, what we need to do is we need to find the range of the function, the original function. If we find the range of the original function, then the range of the original function will be equivalent to the domain of the inverse function, since the inverse swaps the domain and arrange with each other. So the problem tells us that the domain for the original function is between the values of -1 and 1. So what we want to do in order to find the range is we want to plug in the endpoints of this domain into our original function. Starting with -1, if we plug in -1 into the original function, F-1 is equal to 5 multiplied by -1 + 1 divided by 3 multiplied by -1 minus 4. This will simplify to be -4 divided by -7. Which is equivalent to 4 divided by 7. So this is one value for the range. Now if we will repeat this process and plug in 1 into the original function, F1 is equal to 5 multiplied by 1 + 1 divided by 3 multiplied by 1 minus 4. This will simplify to be 6 divided by -1, which is equivalent to -6. So what this tells us that the range of F of X The range of FX is going to be from -6 to 4/7. And what this tells us is that the domain of the inverse function F inverse of X is going to be from -6. To 4 or 4/7. This is going to be the final part of our solution. So our inverse function for the original fx function is 4 x plus 1 divided by 3 x minus 5, and the domain is going to be defined from 6 to 4/7. With that being said, the overall solution to this problem is going to be option C. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

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