Question

Find the value of a and b so that the function is continuous everywhere. Show work algebraically, using the DEFNITION of f(x)= [3x−2,]

[ x2+ax+b,1 [−2x+6,x≤1x>2 ]
you can use a graph to check your answer

Answer 1

To find the values of a and b so that the function is continuous everywhere, we set up equations using the three parts of the function. Solving these equations, we find that there are no values of a and b that make the function continuous everywhere.

Step-by-step explanation:

To find the values of a and b so that the function is continuous everywhere, we need to consider the three cases where the function is defined. For x ≤ 1, the function is -2x + 6. For 1 < x ≤ 2, the function is 1. For x > 2, the function is 3x - 2. To ensure continuity, we need the three pieces of the function to be equal at x = 1 and x = 2.

1. Setting x = 1 gives: -2(1) + 6 = 1. Solving for -2 + 6 = 1, we get -2 = 1 - 6.
2. Setting x = 2 gives: 3(2) - 2 = 1. Solving for 6 - 2 = 1, we get 4 = 1.

Since the equations -2 = 1 - 6 and 4 = 1 are contradictory, there are no values of a and b that make the function continuous everywhere.