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

Question

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

f(x)= {9-x^2/x-3, if x < 3

b, if x=3

ax, if x > 3}

Answer 1

Need to find a y-value that is the same for x<3, x=3, and x>3.

Start by simplifying the top expression: (9-x^2)/x-3 = (3-x)(3+x)/(x-3) = (3-x)(3+x)/(-1)(3-x)

So the 3-x on top cancels with the 3-x on the bottom, and makes the top expression -(3+x). When x=3, y = -6

Therefore we need b=-6

and we need ax=-6 when x=3, so 3a = -6, so a= - 2