Construct a rational function that has a vertical asymptote at x = 3 and a removable discontinuity at x= -2 . Explain how you determined your answer.
To construct a rational function with a vertical asymptote at x = 3, we need a factor of (x - 3) in the denominator.
To also have a removable discontinuity at x = -2, we need a factor of (x + 2) in both the numerator and the denominator, canceling each other out.
Therefore, a rational function that meets these criteria is:
f(x) = (x + 2)/(x - 3)
The factor (x - 3) in the denominator creates a vertical asymptote at x = 3. The factor (x + 2) in the numerator and denominator creates a removable discontinuity at x = -2 because they cancel each other out, leaving the function undefined at x = -2.