Identifying function transformations

so this red curve is the graph of f of X and this blue curve is the graph of G of X and I want to try to express G of X in terms of f of X and so let's see how they're related so we pick any X and we could start right here at the vertex of f of X and we see that at least at that point G of X is exactly one higher than that so G of two I could write this down G of two G of two is equal to is equal to F of 2 F of 2 plus 1 plus 1 let's see if that's true for any X so then we can just sample over here well let's see G of let's see F of 4 is right over here G of 4 is 1 more than that F of F of 6 is right here G of 6 is 1 more than that so it looks like we pick any point over here even though there's a little bit of an optical illusion it looks like they get closer together they do if you look try to find the closest distance between the do but if you look at vertical distance you see that it stays a constant it stays a constant 1 so we can actually generalize this this is true for any X G of X is equal to f of X is equal to f of X plus 1 let's do a few more examples of this so right over here here is f of X in red again and here is G of X and so let's say we picked x equals negative 4 this is f of negative 4 and we say we see G of negative 4 is 2 less than that and we see whatever f of X is G of X no matter where we go no matter what X we pick G of X seems to be exactly 2 less G of X is exactly exactly 2 less so in this case very similar to the other one G of X is going to be equal to f of X but instead of adding we're going to subtract 2 from f of X f of X minus 2 let's do a few more examples so here we have f of X in red again I'll relay belit f of X and here is G of X here is G of X so let's think about it a little bit let pick a let's pick an arbitrary point here let's say we have in red here this is this point right over there is the value of F of three so that our F of negative three I should say this is negative three this is the point negative three F of three so negative three F of three now G hits that same value when X is equal to negative one when X is equal to negative one so let's think about this G G of negative one is equal to F of negative three F of negative three is equal to F of negative three and we could do that with a bunch of points we could see that G of G of zero G of zero which is right there let me do it in a color you can see G of zero is equivalent is equivalent to F of negative two so let me write that down G of zero is equal to F of negative two we could keep doing that we could say G of 1 G of 1 G of 1 which is right over here this is 1 G of 1 is equal to F of negative 1 G of 1 is equal to F of negative 1 so I think you see the pattern here G of whatever is equal to F of is equal to the function evaluated at 2 less than whatever is here so we could say we could say that G of X is equal to F of well it's going to be 2 less than X so f of X minus 2 so this is the relationship G of X is equal to f of X minus 2 it's important to realize here when I did f of X minus 2 here and remember I'm taking update the function is being evaluated this is this is the input X minus 2 is the input when I subtract the two this is shifting the function to the right which is a little bit counterintuitive unless you go through this exercise right over here so G of X is equal to f of X minus 2 if it was f of X plus 2 we would have actually shifted F to the left now let's think about this one this one is seems kind of wacky so first of all G of X it's it almost looks like a mirror image but it looks like it's been flattened out so let's think of it this way let's take the mirror image of what G of X is so I'm going to try my best to take the mirror image of it so let's see it gets to about two there then it gets pretty close to one right over there and then it gets about right over there so if I were to take its mirror image it looks something it looks something like this it's mirror image if I were to reflect it across the across the x axis it looks something like this it looks something like this so this right over here this right over here we would call so if this is negative if this is G of X if this is G of X when we flip it that way this is the negative G of X negative G of X when x equals 4 G of X looks like it's I don't know about negative 3 and 1/2 you take the negative of that you get positive I guess it should be closer to you get positive 3 and a half if you were to take the exact mirror image so that's negative G of X but that still doesn't get us there it looks like we actually have to triple this value for any point and you see it here this this gets to 2 but we need to get to 6 this gets to 1 but we need to get to 3 so it looks like this red graph right over here is 3 times this graph so this is 3 times negative G of X which is equal to negative 3 G of X so here we have f of X is equal to negative 3 times G of X and if we wanted to solve for G of X right G of X in terms of f of X we would write dividing both sides by negative 3 G of X is equal to negative 1/3 f of X negative 1/3 f of X