Question

The graph below shows the function f (x) = StartFraction x minus 3 Over x squared minus 2 x minus 3 EndFraction.

On a coordinate plane, a hyperbola is shown. A curve opens up and to the right in quadrant 1, and another curve opens down and to the left in quadrant 3. A hole is at x = 3. Both curves approach x = negative 1.

Which statement is true?
There is a hole at x = 3 and an asymptote at x = –1.
There is an asymptote at x = –1 and no hole.
There is a hole at x = 3 and no asymptote.
There is an asymptote at x = 3 and a hole at x = –1.

Answer 1

Factor denominator: x^2 − 2x − 3 = (x − 3)(x + 1). So
f(x) = (x − 3)/[(x − 3)(x + 1)] = 1/(x + 1) except at x = 3 where the original is undefined.
Thus x = 3 is a removable hole and x = −1 is a vertical asymptote. The first statement is true.