Final answer:
To find the values of a and b that make f continuous everywhere, we need to ensure that f is continuous at the points where the different pieces of the function intersect. This involves setting up and solving two equations using the left-hand and right-hand limits of f(x) as x approaches 2 and 3 respectively.
Step-by-step explanation:
To find the values of a and b that make f continuous everywhere, we need to ensure that f is continuous at the points where the different pieces of the function intersect. In this case, that's at x = 2 and x = 3. For f to be continuous at x = 2, the left-hand limit of f(x) as x approaches 2 should be equal to the right-hand limit of f(x) as x approaches 2. This gives us the equation:
lim(x→2-) f(x) = lim(x→2+) f(x)
Substituting the function values on the left and right side of x =2:
x²-4/x-2 = ax²-bx+1
Similarly, for f to be continuous at x =3, the left-hand limit of f(x) as x approaches 3 should be equal to the right-hand limit of f(x) as x approaches 3. This gives us the equation:
lim(x→3-) f(x) = lim(x→3+) f(x)
Substituting the function values on the left and right side of x =3:
ax²-bx+1 = 4x-a+b
Solving these two equations simultaneously will give us the values of a and b that make f continuous everywhere.