Find the inverse function (on the given interval, if specified...

Question

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\left(x\right)=8-4x$

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function F of X equals -5 + 7 X, identify its inverse function from the following options, then graph the function and its inverse. Awesome. So it appears for this particular prompt. We're asked to solve for two separate answers. Our first answer is we're trying to identify the inverse function for F of X, so we're trying to find the inverse function of F of X. That's our first answer. And then our second answer that we're trying to solve for is we're trying to create a graph of the original function F of X, and we're also supposed to add. On the same graph, the graph for the function for the inverse, so both F of X and F to the power of -1 of X, so F of X and its inverse function have to be on the second. So the F of X have to be on the graph together, and that is our second answer that we're trying to solve for. So once again we're trying to figure out the inverse function. That's our first step. And then our second step is we're trying to create a graph that has the original function curve and its inverse curve on the same graph. So, with that said, let's read off our multiple choice answers to see what our final answer for the inverse function might be, noting that the inverse function is going to be denoted as F to the power of -1 of X is equal to some expression. So A is 1/7X minus 5/7, B is 1/7X plus 57, C is 1/7X plus 5, and D is 17 X minus 5. OK. So our first step in order to help us solve for this particular problem is we need to, once again, we're trying to figure out the inverse function and to do that we need to start by replacing F of X with Y. And when we do that, we will find that Y is equal to -5 + 7 X. So now our next step that as we should recall, is we need to interchange X and Y. So when we do that, we will find that X is equal to -5 + 7Y. So now we need to rearrange this expression to isolate and solve for Y. and when we do that, using a little bit of algebra, we will find that Y is equal to X + 5, all divided by 7. Fantastic. So, Therefore, the inverse function, which will be F. So it's gonna be F to the power of -1 of X, that's our inverse function notation. So the inverse function is going to be equal to X + 5 divided by 7. Which is also going to be equal to, when we do a little bit of simplifying, we can also write 1/7X plus 5/7. Fantastic. So let us put a box around that because that is our final answer for our inverse function. So this is F to the power of -1. Of X, so F to the power of -1 of X, that is our inverse function. So right now, let's take a moment here to look at our multiple choice answers. So our multiple choice answer that this corresponds to is multiple choice answer letter B. So we've already solved for one answer, but we still have to create our graph. So in order to graph, let us first focus on the following function. We need to graph. F of X. is equal to -5. Plus 7 X because that is our original function that is given to us. So in order to graph this, we must first figure out some coordinate points. So let us plug in some different values for X and Y. So let us plug in for where an X is equal to 0 as our first point. So when X is equal to 0, let us note that. Y is going to be equal to -5 + 7, and then when we plug in 0, it'll be 7 multiplied by 0, which will be equal to -5. And similarly, when Y is equal to 0, so let's make a note here. So now when Y is equal to 0, let us note that it's going to be 0 is equal to -5 + 7 X. And when we do a little bit of simplifying, when we rearrange this expression to isolate and solve for X on one side, we will find that X is equal to 5/7. So without a doubt. We should note Now we need to plot the following coordinate points because we're dealing with a line equation or a line for a function for a line curve. So with that in mind, we need to make sure we plot points at 0, -5, noting that this is our XY coordinate plane layout. This is our coordinate I said coordinate plane, I meant coordinate point setup. So it's gonna be 0.5, and we also need to plot a point at 5/7. 0, and we need to connect these two points with a line going through these two coordinate points. Awesome. So that is how we're going to, we're going to use these two coordinate like these two coordinate points which I've written and read, and we're going to use those to plot our first curve, which is for the original function of FF X. So now in order to graph our inverse function, which, as we should recall, our inverse function is F the power negative one. Of X is equal to and let's just use our one form, which is X + 5 divided by 7, which when we write this as Y is equal to when we do in our simplified form. 1 divided by 7 or 1/7X plus 5 7, which as you can see, that's in slope intercept form. That's our that's the slope that's the equation for a function of a line. we have MX plus B formatting. Which is good, so that means we're gonna have another line in our graph. So with that said, We need to note once again that we need to figure out different points on our line here because we need to find two separate coordinate points in order to connect them to draw our line properly on our graph. So like we did for our original function, we just need to plug in two different, we need to figure out two coordinate points. So let's plug in X equals 0 into our equation for our inverse function. So once again, so this is X equals 0. So when X equals 0, we could say that Y is equal to 1/7, and then we need to multiply it by 0, and then we need to add 5/7s, which will mean when we isolate and solve for Y that Y will be equal to simply 5/7. Awesome. And when Y equals, once again, 0 for this case, we can plug in our value for 0. We'll get that 0 is equal to 1/7 X, and then we need to add 5 cents, and then once again when we rearrange this equation to isolate and solve for X all by itself. We will find that X is going to be equal to -5. So I skipped some steps, but essentially all we're doing is we're taking our expression, which is 1/7X plus 5 cents, which is equal to 0, and we're going to be rearranging it to isolate and solve for X. And when we do that, we will find that X is equal to -5. So as a result, because of These two points, we need to note that we need to plot the following coordinate points. We're gonna be plotting 0.com. 57, and we also need to we need to plot the point -5.0. So those are our two coordinate points that we need to connect and draw a line through. So the line is going to be connecting these two points, and that is denoting our second. Curve, which is for our inverse function. So, with that said, now that we have all the information needed that we that we'll ultimately need to use to help us plot our functions into one graph, we should get the following graphs. So I've gone ahead and plugged in our original function and our inverse function. Into our or into the graphing, so I plugged it into the graphing software of my choice and as you could see for our first function, so I've done them separately. So our first graph, which is on the far left, this is for our Original function. So this is for once again for when Y is equal to F of X. So this is our original function. And then our second graph going from the left to the right is our inverse function. So this is when Y is equal to F the power of -1 X. And then finally our Last graph, which is on the far right, which I'll put a green box around it. This is going to be our final answer. This is the final answer that we need for our completed graph, which has both of our curves for the original function and for our inverse function. And as you can see, For when we graphed them separately, we could see that we plotted our two points, so we have it at 0.5, and then we have it once again at 5/70, and the line passes through both of those points. And then we also have for our inverse function, we have the point at -5.0. And then we have it at 0, and then it goes up to 5/7. Awesome, and that's it. And then when you combine both of them together on the same graph, they will intersect each other. At it looks like about, so it's not quite one, so it's gonna be. Like 0.8 something or something. I'm not, don't, don't quote me, it's just an estimation, but they intersect here at this green. I drew a little green dot here at their intersection point, which for this particular problem, it's not valid. We don't have to really worry about it, but we can see from our graph that these two curves will intersect each other. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.
$f\left(x\right)=8-4x$

Key Concepts

Inverse Functions

Graphing Functions

Symmetry in Graphs

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