Calculus Question

Question

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

f(x) =	x2 − 4
x − 2
	if x < 2
ax2 − bx + 3		if 2 ≤ x < 3
2x − a + b		if x ≥ 3

a= ?

b= ?

Answer 1

to make the function continuous, you need to make sure that at each boundary of the ranges, so when the piece wise function changes, that the y is still the same, that means that for example, at x = 3, ax² - bx + 3 = 2x - a + b. This is because if you graph these two "mini functions" they must meet at the same point. if the y-value for x=3 of both functions is different, the graph will have a gap.

you need to solve for a and b so you have two variables and need two equations.

you need to ensure continuity at two points where the middle range stops and starts: 2, 3

to set continuity at x = 2

x - 2 = ax² - bx + 3

which is re written as

2 - 2 = a(2)² - b*2 + 3

to set continuity at x = 3

ax² - bx + 3 = 2x - a + b

which is re written as

a(3)² - b*3 + 3 = 2*3 - a + b

solving for a and b with simple algebra leads to

a = 9/2

b = 21/2

(note: due to the formating it's unclear which function is for x < 2. I read it as "x - 2" but if that isn't what the question says, replace "x - 2" with the actual function for x < 2)