 Welcome back everyone. In this video, we're going to talk some more about finding the inverse of a function algebraically And we're going to look at some examples involving square root functions So let's find algebraically the inverse of the function f of x equals the square root of x minus 2 We're also going to find the domain and range of this function here And so I do want to mention that real quick if we mentioned the domain and range of f The graph of the function is going to look something like the following This will be this function right here is our standard square root function But it's been shifted to the left by 2 because of the x minus 2 inside of the square root and as such our graph would look something like this again It looks like the standard square root function, but shifted to the right by 2 therefore the domain of f is going to equal 2 to infinity and Then the range of f because we didn't do any vertical transformations the range will just be 0 to infinity So we can look at the domain and range of this function very quickly through our graph transformations What we can already do is we can already predict what the graph of f inverse is going to look like if we draw the diagonal line Y equals x we know that the graph of f inverse will be the reflection of This fun of the original function f across this diagonal line So we're going to get something like the following F inverse right there. That's what this graph would look like Remember the white graph here was f So let's consider the algebraic function for f inverse, right? So it seemed previously that the function f is given by the relationship y equals the square root of x minus 2 Where we replaced f of x with y right here in order to go to the bizarro realm and get f inverse We're going to switch the roles of x and y right here y becomes an x and x becomes a y and now we have to solve for the Inverse function have to solve for y in this case by performing the inverse operations We're trying to get y all by itself on the That's on the right-hand side of the equation. Well to get rid of the square root function We apply its inverse x we square both sides This gives us that y minus 2 is equal to x squared and then we'll add 2 to both sides We add 2 to cancel the plus 2 that's on the left hand side. This gives us y equals x squared plus 2 But when we'd say y here, we're really going to put in the name of the inverse function f inverse So we're very specific. So here's the formula for inverse function F inverse of x is x squared plus 2 But we do have to be careful about domain and range because of the domain convention Unless we specify otherwise one when they see the formula y equals x squared plus 2 You would assume that the domain is maximal. That is it be all real numbers But because of the domain and range of the original function in order to make the inverse function one to one We actually don't get the left hand side of this parabola Instead we're going to switch the domain and range of f to give us the domain and range of f inverse The the domain of f inverse is going to be the range of f Which means it's going to be zero to infinity Which we see right here you get any x value to the right of the y-axis there And the range of f inverse it's going to be the domain of f Which is 2 to infinity and looking at the graph here You're only going to get those y coordinates which are greater than Or equal to 2 And so the range and domain of f inverse are just going to be the opposite of the range and domain of f The domain of f becomes the range of f inverse and the range of f becomes the domain of f inverse Let's look at another example quite similar in nature We have a function f of x which equals 2 plus the square root of x minus 4 So the formula that defines f is going to be y equals 2 plus the square root of x minus 4 Switching to the inverse function the equation that will define f inverse will be created by swapping the roles of x and y The y is going to become an x and the Let's see the x will become a y and so now we just proceed to solve this by doing all the inverse operations So track 2 from both sides so the two cancels on the left hand side I like to put the y on the left hand side, so I'm going to switch the order here We get the square root of y minus 4. This is equal to x minus 2 Next I want to square both sides make sure that you're squaring the entire expression x minus 2 Like so this would then give us y minus 4 on the left is equal to x minus 2 squared For which if we want to we can we could multiply that I'm going to leave it factored For reasons that will become clear in just a second and then minus I guess we want a plus 4 to both sides and this is going to give us that y Equals x minus 2 squared plus 4 But then again, I'm going to get rid of the y in the final stage And come over here and say f inverse of x equals that So it's very clear that this formula is giving us the formula of the inverse function If we were to compose this function with f their composition would just be x these operations will cancel each other out In terms of graphing these things We could we could sketch a graph of this very very quickly Some things to note here. I'm going to put a little bit of tick marks on the On the axes there Something like that. So the original function right here, you can see that this is the standard square root function It's been shifted to the right by four. It's been shifted up by one Um, and so if we count that out, you're going to get 1 2 3 4 to the right and then 1 2 up your graph will look something like this In particular there's this point 4 comma 2 Whoops 4 comma 2 and this is the graph for f Now we'll draw the diagonal line in there to emphasize the symmetry that's going to happen If we graph the function for f inverse This function right here is going to be the standard parabola It's been shifted to the left by 2 and it's been shifted up by 4 so we go 1 2 to the left or to the right excuse me and then we go 1 2 3 4 Upward and then it's going to look like the left side of a parabola And so this would be the graph of f inverse like so it'll emanate from this point to comma 4 They're going to be reflections of each other And so we can very quickly see that the domain of the square root function We can find that out by solving the inequality x minus 4 is greatly equal to 0 Or we can see this from the graph the standard graph has been shifted to the right by four units So it's the domain of f would be 4 to infinity And in the range this one's a little bit harder to do algebraically But we can see geometrically that you took the standard square root and you moved it up by 2 That changes the range to be 2 to infinity And so then considering the domain and range of f inverse the domain of f inverse is going to equal The range of f which is 2 to infinity as you can see here We get all x coordinates to the right of 2 And then the range of f inverse This is going to equal the domain of f Which is 4 to infinity and you can see we get everything above 4 on the y-axis So this is how we can compute the formula for functions inverse algebraically Just take the original equations swap x and y and then solve for y We can find the domain and range of those functions by computing the domain and range of the original function Probably going to do that using transformations And then by switching the rules of domain and range you get the domain and range of f inverse One thing I want to mention here is the original graph was shifted to the right by 4 and up to On the other hand, this function was shifted to the right by 2 And shifted up by 4 The horizontal shift of the original graph becomes a vertical shift in the inverse because When you take the inverse function the horizontal becomes the vertical On the other hand the original vertical shift of 2 Turns into a horizontal shift of 2 when you switch to the inverse function Because the vertical will become horizontal when you switch to the inverse function Because the inverse function by its nature it swaps the roles of x and y Everything horizontal becomes vertical and vertical for horizontal The inverse function is just the original function stating on its head so to speak