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CCSS.Math:

let f of X equal 3 to the negative X minus 5 and G of x equals negative x to the third minus 4 x squared plus X plus 6 the graphs of y equals f of X and y equals G of X are shown below and we see them right over here this y equals f of X is in that is in that purplish color let me see if I can I can get that same purplish color so that is in that so let me underline it now so f of X is in that purplish color so y equals f of X that's right over here in purple and then y equals G of X that is in blue so y equals G of X G of X is defined right over there y equals G of X well that is graphed in blue and we see that they intersect at the point a comma B so there's a couple of ways there's a couple of ways to think about this we could say that when X is equal to a f of X and G of X equal each other or we could say f of a and this is coming from this point of intersection let me draw a little arrow here so our big arrow that point of intersection lets us know that F of a F of a is equal to G of a is equal to G of a which is equal to B G of a which is equal to B they both if you input a into the function f you're going to get B if you input a into the function G you're going to get B and so the point a comma B is on both graphs both y equals G of X and y equals f of X and so from here you can make some interesting you could make some interesting statements for example you could just say well what is f of a f of a is 3 to the negative a power minus 5 is going to be equal to what's G of a G of a is negative a to the third minus 4 a squared plus a plus 6 so you could say this and that would be equal to that either of those would be equal to B all right I think we've analyzed that a good bit so now let's actually answer their questions and normally I'd suggest that you look at the questions before you actually try to solve it but I just wanted to do this just to really squeeze out as much as we could out of the information they gave us so they tell us the value x equals a is a solution to which of the following equations select all that apply so this first one is 3 to the negative X minus 5 is equal to B well we already know that we already know that 3 to the negative a minus 5 3 to the negative a minus 5 is going to be equal to B is going to be equal to B this over here this is equivalent to saying this is equivalent to saying f of X f of X is equal to B and we know f of X equals B when x equals a when x equals a f of X is equal to B this expression is equal to B so we know that that first one is true now the second one is just saying f of X is equal to G of X well we know that when X is equal to a f of X is equal to G of X that f of a is equal to G of a because as a reminder this right over here is our definition for f of X and this over here is our definition of G of X so this is just saying f of X equals G of X when does that happen well that happens when x equals a we already saw it up your F of a is equal to G of a they both equal B and so both of these are going to be equal to each other when X is equal to a so I will check that one as well