Finding constants a and b for a continuous piecewise function

[SmittenWCalc]

Homework Statement

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.

Homework Equations

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.

 

[Mark44]

Homework Statement

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.

Homework Equations

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 = 3Now, 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.

 

[SammyS]

Homework Statement

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.

Homework Equations

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 .