What value of a will make the following function continuous? f(x)={2x−3, x≤2x2−ax+1, x>2 (1 point) Responses −2 − 2 −1 − 1 0 0 1 1 2
To make the function continuous, we need to ensure that the left-hand limit and the right-hand limit are equal at x = 2.
The left-hand limit is the value of the function approaching x = 2 from the left side, so we can substitute x = 2 into the first part of the piecewise function:
f(2) = 2(2) - 3 = 4 - 3 = 1
The right-hand limit is the value of the function approaching x = 2 from the right side, so we can substitute x = 2 into the second part of the piecewise function:
f(2) = 2(2)^2 - a(2) + 1 = 8 - 2a + 1 = 9 - 2a
To make the function continuous at x = 2, we need the left-hand limit and the right-hand limit to be equal:
1 = 9 - 2a
Simplifying the equation:
2a = 8
a = 4
Therefore, the value of a that makes the function continuous is 4.