In Exercises 95–96, find all values of x satisfying the given con...

Question

In Exercises 95–96, find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, and (ƒ o g) (x) = 7.

Accepted Answer

Hello today we're going to be solving for the values of X that satisfy the three functions given to us. Now. The first function is F of X is equal to three X minus two. The second function is G of X is equal to X squared minus five. X plus six. And our third function is F of G of X equal to four. Now in order to start solving for X, we're gonna go ahead and let F of G of X be our starting point. And the reason for this is because we're given a value for F of G of X. We're told that it's equal to four. So we can go ahead and use that as a starting point to figure out what X is now. In order to figure this out, we need to first find out what F of G of X is now F of G of X can also be rewritten as F with the function of G of X inside of it. So what this means is we are taking our function G of X and we're plugging it into every X inside of our function F of X. Doing so is going to give us three times X squared minus five X plus six minus two. Now, once again, all I did was take the value of G of X and replace it for X in F of X. Alright now I need to go ahead and simplify this in order to simplify this, I'm just gonna go ahead and distribute three into X squared minus five X plus six. Doing so is going to give me three X squared minus 15 X plus 18 minus two and 18 minus two is going to give me 16. So I have three X squared minus 15 X plus 16. Now this is gonna be the function of F of G of X. But keep in mind we are told that this function is equal to four so we can go ahead and set this equal to four. Now we have three X squared minus 15, X plus 16 equal to four. But since we are solving for X, we want to go ahead and set this equation equal to zero in order to do so we just need to go ahead and subtract four from both sides of this equation and doing so is going to give me three X squared minus 15, X plus 12 equal to zero. Now this is in the form of a quadratic because we can solve this using factory so we can use the quadratic equation which is X is equal to negative B plus or minus the square root of B squared minus four. A. C all over two. A. Now we need to identify what are a. B and C. R. A is just the coefficient in front of our X square term. So A is equal to three. B is going to be the coefficient in front of our X term. So B is equal to negative 15 and the sea is just going to be the constant at the end of the trine Amiel, so C is equal to 12. If we plug this into the quadratic equation we get X is equal to negative negative B which is 15 plus or minus the square root of negative 15 squared minus four times A. Which is three times C, which is 12. All of that over two times A which is going to be two times three. Now we need to go ahead and simplify this. So writing it over here, X is equal to 15 plus or minus the square root of negative 15 squared which is minus four times three times 12, which is 144 and all of that is going to be over two times three which is six now 225 minus 144 is going to give us 81 so X is equal to 15 plus or minus the square root of 81/6. Now the square root of 81 can be evaluated to nine, so X is equal to 15 plus or minus 9/6. Now this is saying that there are two values for X. And those two values are going to be when X is equal to 15 plus 9/6 and when X is equal to 15 minus 9/6 now 15 plus nine is going to give us 24 so X is going to equal to 24/6 and 24/6 can be reduced to four, so X is equal to four and this is going to be our first solution for X. Now, we need to go ahead and solve for the other solution of X, while 15 minus nine is equal to six, so X is going to equal to 6/6 and 6/6 can be reduced to one, so X is equal to one. And this is gonna be our second solution for X. So the two values of X that satisfy the three functions given to us are going to be four and one, and that means that the answer to this problem is going to be B. So I hope this video helps you in understanding how to solve the values of X and I'll go ahead and see you all in the next video.

In Exercises 95–96, find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, and (ƒ o g) (x) = 7.

Key Concepts

Function Composition

Solving Equations

Quadratic Functions

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