Finding inverse functions: linear

so we have f of X is equal to negative x plus 4 and it's f of X is graphed right here on our coordinate plane let's try to figure out what the inverse of f is and to figure out the inverse what I like to do is I set Y I set the variable y equal to f of X or we could write that Y is equal to negative x plus 4 right now we've solved for Y in terms of X to solve for the inverse we do the opposite we solve for X in terms of Y so let's subtract 4 from both sides you get Y minus 4 is equal to negative x and then to solve for x we could multiply both sides of this equation times negative 1 and so you get negative y plus 4 is equal to X or just says we're always used to writing the dependent variable on the left hand side we could rewrite this as x is equal to negative y plus 4 or another way to write it is we could say that F inverse F inverse of Y is equal to negative y plus 4 so we this is the inverse function right here and we've written it as a function of Y but we can just rename the Y as X so it's a function of X so let's do that so if we just rename this y is X we get F inverse of X is equal to negative x plus 4 these two functions are identical here we just use Y is the independent variable or is the input variable here we just use X for they're identical functions now just out of interest let's graph the inverse function and see how it might relate to this one right over here so if you look at it it actually looks fairly identical it's negative x plus 4 it's the exact same function so let's see if we have the y intercept is 4 it's going to be the exact same thing the function is its own inverse is its own inverse so if we were to graph it we would put it right on top of this right on top of this and so there's a couple of ways to think about it in the first inverse function video I talked about how a function and their inverse they are the reflection over the line y equals x so where is the line y equals x here well aligned y equals x looks like this line y equals x looks like this and negative X plus 4 is actually perpendicular to Y is equal to X so when you reflect it you're just kind of flipping it over but it's going to be the same line it is its own reflection and let's make sure that that actually makes sense when we're dealing with when we're dealing with I guess this is the standard function right there if you input a 2 it gets mapped to a 2 if you input a 4 it gets mapped to 0 what happens if you go the other way if you input a 2 well 2 to get some app 2 to either way so that makes sense for the regular function 4 gets mapped to 0 for the inverse function 0 gets mapped to 4 so it actually makes complete sense let's think about it another way for the regular function let me write it explicitly down this might be obvious to you but just in case it's not it might be helpful for so f of let's pick F of 5 f of 5 is equal to negative 1 or we could say the function f Maps us from 5 to negative 1 now what does what does F inverse do what's F inverse F inverse of negative 1 F inverse of negative 1 is 5 is equal to 5 or we could say that F Maps us from negative 1 to 5 so once again if you think about kind of the sets there are domains and our ranges so let's say that this is the domain of F this is the range of F F will take us from F will take us from 5 5 to negative 1 that's what the function f does and we see that F inverse takes us back from negative 1 to 5 F inverse takes us back from negative 1 to 5 just like just like it's supposed to do let's do one more of these so here I have G of X is equal to negative 2x minus 1 so just like the last problem I like to just set y equal to this so we say Y is equal to G of X which is equal to negative 2x minus 1 now we just solve for X y plus 1 is equal to negative 2x just added 1 to both sides now we can divide both sides of this equation by negative 2 and so you get negative Y over 2 minus 1/2 is equal to X or we could write X is equal to negative Y over 2 minus 1/2 or we could write F inverse as a function of Y is equal to negative Y over 2 minus 1/2 or we could just rename Y as X and we could say that F inverse F inverse of oh maybe careful here that shouldn't be an F the original function was G so let me be clear that is G inverse of Y G inverse of Y is equal to negative Y over 2 minus 1/2 this we started with the G of X on an f of X make sure we get our notation right or we could just rename the Y and say G inverse of X is equal to negative x over 2 minus 1/2 now let's graph it it's y-intercept is negative 1/2 so it's right over there and it has a slope of negative 1/2 it has a slope of negative 1/2 let's see if we start at negative 1/2 if we move over to 1 in the positive direction we'll go down 1/2 if we move over 1 again we'll go down 1/2 again if we move back with so we'll go like that so the line I'll try my best I'll try my best to draw it will look something like that it will just keep going so it'll look something like that I'll keep going in both directions and now let's see if this really is a reflection over y equals X y equal X y equals x looks like looks like that and you can see they are reflections if you reflect this guy if you reflect this blue line it becomes this orange line but the general idea you literally just the function is originally expressed is solved for Y in terms of X you just do some algebra solve for X in terms of Y and that's essentially your inverse function as a function of Y but then you can rename it as a function of X