Question
2. The function f is defined by f(x) =x +x -5 and f(1) =_ 3 . If f is the inverse function of f find the value of 'Y(-3) _
2. The function f is defined by f(x) =x +x -5 and f(1) =_ 3 . If f is the inverse function of f find the value of 'Y(-3) _
Answers
Find the inverse function of $f$. $$f(x)=\frac{3 x+2}{2 x-5}$$
Direct. So, um, we know that f of X is equal to five minus two x, and we are looking for F inverse of three. So what kind of helps me whenever it's set in this function kind of notation situation? What I like to do is I kind of like to switch things around, and this is a silly little game I like to play. Um, normally, they would say the inverse is that you're solving for X with the textbook. That's what it says for myself. I kind of like to think about I'm solving for X to find my inverse. So I know my inverse function of eggs and this is going to be my new X over here. So I kind of take this f this function and I switch it over with my ex and everything else kind of stays the same on the first round. And then I saw. So I'm going to start this off by subtracting five and subtracting five. So now I have X minus five is equal to two times Excuse me negative two times the function in verse of X. So I'm going to divide by negative too, and divide by negative, too. And while law now I know the function in verse of X is equal to X minus five, all divided by negative too. So now I need to know the function in verse when it's three. So if I did that, I would say, is equal to three minus five divided by negative too. Um, in the numerator three minus five is equal to negative two divided by negative too. And therefore the function inverse of three is equal to one. So here's how you would check your work if I were to take this one and put it into my regular function my f of X that I originally started with over here, the answer should be three. So let's check that up. So f of at Sorry, not ex anymore. I was going to use X. But let's remember that what we're doing is we're actually finding f of one is equal to five and then minus two times one. And so now that is equal to three. So therefore, this is how you check your work and it checks out. So if the function in verse of three is equal to one
All right, We're given this function of X equals X squared plus two and were asked to find the inverse. It's the first thing to recall is that f of X is simply why and what it means to find the inverse is just fine. What happened? What's the value of our function if we switch X and why? So the first thing we're gonna do is replace why, with X and X and Y in our function, simply rewriting each accident. Why and each wise and X now to find the inverse of the function we need to now, Saul, for our new Why So we got to get why by itself it's a simple two step equation. First thing we need to do a subtract two from both sides that eliminates 20 On the left, we have X minus two, and on the right, we have five. Why? And now to finish simplifying to divide by our coefficient and why this becomes one divide by five and we get our final answer is that why equals X minus two divided by five. That is the inverse function from our leverage original function F of X
Way have the function f of X equals two minus X cute to the fifth. And we want to find the inverse Think of up of X is why interchange X and y insult For wise, we have to find or take the fifth root of each side. First, we have the fifth root of X equals two months. Why cubed? Then subtract two to both sides and then, uh, we can divide everything by negative one we have. Why Cube equals this would just change. The science would be negative and a positive, which have just been a right as positive two and then the negative fit through acts of to minus the fifth root of acts. And then we would cube root both sides get rid of the cube. So we've got our inverse function of X would be the cube root of two minus the fifth root of pets.
All right, So let's do a review question for an versus If I have the function F of ax equals five X over X plus two and we'll set that equal to lie. The typical format is too. Switch your variables and because we know the original function, you input X and output. Why? And on the inverse you'll be in putting why, and out putting X. So we're going to change that to a five y over Y Whoops. Why plus two equals X. And now we need to solve or why. So keep that in mind. You're gonna get all the wise on one side. Now I have to start off by multiplying both sides of the equation by wyffels, too. And then this factor cancels that factor. Now, you might think you like this in factored form, but we actually don't. We want to distribute it now. Remember what we're doing. We're solving for why, so I need to get the y terms together, and I could get this even though it has an ex attached. I need to subtract ex wife on both sides Now. I want to pull out that common factor of why and I have one more step. Dubai those sides by five minus X and I have two X over five, minus fax. And if I want to write that in functional notation, I need know that this is the inverse with the input variable being acts. Can I have two acts over five minus ax?