# Find the values of a and b that make f continuous everywhere

by SmittenWCalc
Tags: continuous, values
 P: 1 1. The problem statement, all variables and given/known data This is a piece wise function of course. f(x) = (x2-4) / (x-2) if x is less than two. ax2 - bx + 1 if x is greater than or equal to 2, or less than three. 4x - a + b if x is greater than or equal to three. 2. Relevant equations 3. The attempt at a solution Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting 4 = a4 - b2 + 1 but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get 3 = a4 - 2b Now I am definitely stuck.
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 Quote by SmittenWCalc 1. The problem statement, all variables and given/known data This is a piece wise function of course. f(x) = (x2-4) / (x-2) if x is less than two. ax2 - bx + 1 if x is greater than or equal to 2, or less than three. 4x - a + b if x is greater than or equal to three. 2. Relevant equations 3. The attempt at a solution Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting 4 = a4 - b2 + 1 but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get 3 = a4 - 2b Now I am definitely stuck.
Written in a more useful way, your equation is

4a - 2b = 3

Now, what about at x = 3? You want the function to be continuous there, as well, right? What needs to happen for f to be continuous at x = 3?

That should give you another equation so that you have a system of two equations in the unknowns a and b.
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 Quote by SmittenWCalc 1. The problem statement, all variables and given/known data This is a piece wise function of course. f(x) = (x2-4) / (x-2) if x is less than two. ax2 - bx + 1 if x is greater than or equal to 2, or less than three. 4x - a + b if x is greater than or equal to three. 2. Relevant equations 3. The attempt at a solution Alright, I know enough to factor the top of the fist equation and get x+2. That means when x is two, f(x) is four. We can use f(x) in this case because we are making the function continuous. I've gotten as far as plugging in this value in the second equation and getting 4 = a4 - b2 + 1 but I don't know what to do from here, or how to get the values of a and b. I think I subtract one from the right and get 3 = a4 - 2b Now I am definitely stuck.
Hello SmittenWCalc. Welcome to PF!

So, you have the following:
$\displaystyle \lim_{x\to\,2-}f(x)=\lim_{x\to\,2-}(x+2)=4\ .$

$\displaystyle \lim_{x\to\,2+}f(x)=\lim_{x\to\,2+} (ax^2+bx+1)=4a+2b+1\ .$
Do something similar at x=3 .

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