In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ...

Question

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of f and ƒ¯¹. f(x)=2x-1

Accepted Answer

Welcome back. I am so glad you're here. We are given this function F of X, which is equal to four X minus eight. And were asked to do several things here. We need to find the equation for its inverse, which will be f inverse of X equals something. We'll figure that out shortly. Next we will graph both the function and its inverse. I'm going to draw a rough sketch for a graph. We've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. And then finally, we will provide the domain and range in interval notation for both functions. All right, let's get started with the equation for the functions inverse. I'm going to rewrite the function down here. That is F of X which equals four x minus eight. Next I'm going to replace F of X with Y. So we've got Y equals four x minus eight. And now we are going to do some switching here wherever we see a Y. We're going to replace that with an X. So instead of Y equals, it's going to be X equals. And wherever we see an X, we're going to replace that with a Y. So this will be four Y instead of four X still minus eight. And now that y is just miserable that it is surrounded by all of these other numbers and it wants to be all by itself. That is the only way it will be happy. So we need to do the work to get the Y by itself again. So let's add eight to both sides of the equation. So on the left will have X plus eight. That will equal four. Y on the right. Why is almost by itself? It's almost happy. We can divide both sides by four. And now why is just thrilled? It is by itself on the right hand side. And now we've got a couple of options because the answer choice writes it this way, I'm going to put X plus eight divided by four as our equation for f inverse of X. But we can also simplify this a little bit and it will be helpful in a few minutes. We can divide both of the numbers in the numerator, the X and the eight by the four in the denominator. So that would be X divided by four, which is the same as 1/4 X Plus eight divided by four which is two will use that equation in a few minutes. Now let's work to plot our points to graph this F of X. Okay, so bringing it back up here, normally we would set up a table, we would choose some X values and then we would plug those X values into the original function. Figure out the corresponding y values and then plot those points. However, as you may recall from previous lessons, this function is in the format of a line and specifically it's in the format of Y equals M X plus B. The slope Y intercept format of a line where M. Is our slope and B. Is our Y intercept with those two pieces of information. We can graph this function the line and the function. So for our slope that's immediately preceding the X. So our slope is four or 4/ for our Y intercept. That's here at the end, that's the B. R Y intercept is negative eight. So looking at our coordinate plane, we can start at the origin 00 in the middle and we're going neither to the left nor to the right for our X values. Because we're talking about a Y intercept and we will go down eight units on the Y axis to approximately here and this is the 0.0 negative eight and there's our Y intercept. Now we know our slope, our slope is 4/1. That's our rise over our run. So our rise tells us the change in the Y values will go up the Y axis to negative four. And that are run tells us the change in the X values will go to the right one unit that's approximately here at one negative four. For the next point we'll plot one more point and then draw our line. So from one negative four, we're again going to go up four units for our Y value. Now it's at zero for our Y value and to the right one for our X value. So that's here at 20. Now we can draw this line and there is our F of X. Alright, we've got one of them plotted. Now let's work on graphing the inverse. Now very similarly we could start figuring out some X and Y values. But this inverse is also in the form of an equation of a line. And I'm going to use this other formula that we have down here. This 1/4 X plus two equals Y. And that is in Y equals mx plus B. I just have the Y. On the right hand side of the equation which is fine. So our slope our m immediately in front of the X. That is 1/4. That's our rise over run and our Y intercept R. B is positive two. So now we can look at our coordinate plane starting at the origin 00 in the middle. We will go up two units for our Y value. We're not moving to the right or to the left for our X value. So I'll label this as zero to and we know that our slope is 1/4. So we're going to go up one unit for the Y. Value from zero to and to the right four units for our X value. So that's approximately here. I will label that that will be at four. Three. We can connect these points, draw our line, this is f inverse of X. Alright. We've done the equation, we've done the graphing. Now we need to figure out the domain and the range. Well, the domain is going to be all of the possible X values. So looking first at our F of X, all of the possible X values. Well, it's a straight line, it's all of the X values. So it's going from negative infinity all the way up to positive infinity. We express that in interval notation, starting with a parentheses, the negative infinity comma positive infinity. And ending with the parentheses, the range is all of the possible Y values for our function. And again, it's a straight line that is going to be everything from negative infinity to positive infinity. So we can state that again negative infinity to positive infinity. Now for the inverse function, the domain and the range again there it's a straight line. We're going from negative infinity to positive infinity for both the domain and the range. Okay, we have got all of the things figured out. Now let's figure out which one is our correct answer choice. Looking at our graph, we want to look for an F of X that has an X intercept at 20. Looking at answer choice A. They have an F of X. With an ex intercepted 20. We're looking pretty good. How about our f inverse of X? They have a Y intercept at 02 and look at this answer choice A F inverse of X has a Y intercept ID 02. We're looking pretty good. Now let's compare the rest of the information they have. F inverse of X equals X plus eight, divided by four. We have F inverse of X equals X plus eight, divided by four. Great. They say that the domain of both function and the function and its inverse are from negative infinity to infinity. And yes, we said that to the domain is from negative infinity to positive infinity. How about the range the range of both the function and its inverse from negative infinity to positive infinity. And yes, that is what we said. Answer choice A. You work alright, Well done. We'll catch you on the next one.

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of f and ƒ¯¹. f(x)=2x-1

Key Concepts

Inverse Functions

Graphing Functions

Domain and Range

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