This is extremely hard to read. Is your function ?
f(x) =
x^{2} − 4 x − 2 if x < 2 ax^{2} − bx + 3 if 2 ≤ x < 3 4x − a + b if x ≥ 3
OK guys so I'm having trouble with this, it includes a lot of algebra steps but I'll put some of them to show that I DID attempt it several times, not sure if I'm just tired or I messed up at some point....
I factored out at the start of coarse...
(x-2)(x+2)/ (x-2) = ax^2 - bx + 3 -----> (2+2) = a(2)^2-b(2)+3
4 = 4a - 2b +3
4 - 3 = 4a - 2b
1 = 4a - 2b
1+2b/4 = 4a/4
1/4+1/2b = a
ax^2 -bx + 3 = 4x - ab
a(3)^2 - b(3) +3 = 4(3) - a + b
9a + a - 3b - b = 12- 3
10a - 4b = 9
10(1/4 + 1/2b) - 4b = 9
5/2 + 5b - 4b = 9
5/2 - 1b = 9
-1b = 13/2
= -13/2
1/4 + 1/2(-13/2) = a
1/4 + (-13/4) = a
- 3 = a
a = -3 b = -13/2
It should be clear that the function needs to approach 4 if you make x approach 2 from the left (why). For the function to be continuous at x = 2, it needs to approach the same value from the right. So that means .
For the function to be continuous at x = 3, we require the function to approach the same value from the left as from the right. So .
Simplify both of these equations and solve them simultaneously for a and b.