Graphs of nonlinear piecewise functions

select the piecewise function whose graph is shown below and where I guess we should say to the right I copied and pasted it so it's on the right now so we have this piecewise continuous function so it's not defined for X being negative to or lower but then starting at X greater than negative 2 it starts being defined it's continuous all the way until we get to the point x equals 2 and then it we have a discontinuity and then it starts getting defined again down here and then it is continuous for a little while all the way and then after X is greater than sit when X is greater than 6 it's once again undefined so let's think about which of these functions describe this one over here so this one looks like a a radical function shifted so square root of let's see square root of x would look something like square root of x would look like this let me do it in a color that you could see square root of x would look like this square root of so square root of X looks like this and this just looks like square root of x shifted two to the left so this looks like square root of x plus two this one right over here looks like square root of x plus two and you could verify that when X is negative two negative two plus two square root of that is going to be zero and it's it's not defined there but we see that if we were to continue it would have been defined there let's try some other points when X is negative one negative one plus two is one principal root of one is one let's try two when X is equal to two two plus two is four principal root of four is positive two so this this looks like a pretty good candidate so it looks like our function so let's see let's see it looks like our function it would be and I'm not going to call it anything because it could be phg or f but our function if I were to write it out it looks like over this first interval so it looks like it's the square root of x + 2 4 negative 2 being less than X it's not defined at negative 2 but as long as X is greater than negative 2 and X is less than or equal to 2 and X is less than or equal to 2 so that's this part this part of the function and then it jumps down and now this looks like X to the third X to the third looks something like this so X to the third power looks something something like something like that so let's see negative two to the third power is negative eight so it looks something like that that's what X to the third looks like so 2 to the third power is 8 2 to the third power is 8 so X to the third looks something like that this looks like X to the third shifted over four so I would I'm guessing that this is X minus 4 to the third power and but we can verify that when X is equal to 4 4 minus 4 to the third power we do indeed get the value of this expression being equal to 0 when X is 6 6 minus 2 to 6 minus 4 is 2 to the third power is indeed 8 when X is 2 2 minus 2 2 minus 4 is negative 2 to the third power is indeed negative 8 so this over this over this interval it is X minus 4 to the third power so we could say we could say so this part of the function this part of the function right over here we could say is X minus 4 to the third power 4 for X or X being so for X being greater than 2 or we could say 2 is less than X and it's defined all the way to s X being equal to 6 but not being greater than so X is less than or equal to 6 so which of these choices are what we just put in so square root of x plus 2 well that's not C square root of x plus 2 for negative 2 is less than X is less than or equal to 2x minus 4 to the third for 2 is less than X which is less than or equal six so I would go with that one