Identify any holes in the graph of the rational function: $$ \frac{x^{2}+x-20}{x^{2}-6 x+8} $$

Question

UpStudy AI · Accepted Answer

To identify holes in the graph of the rational function (x² + x - 20) / (x² - 6x + 8), start by factoring both the numerator and the denominator.

1. Factor the numerator:
 x² + x - 20 factors into (x + 5)(x - 4).

2. Factor the denominator:
 x² - 6x + 8 factors into (x - 2)(x - 4).

Now the function is:
 [(x + 5)(x - 4)] / [(x - 2)(x - 4)].

Notice that (x - 4) is a common factor in both the numerator and the denominator. Cancelling this factor gives:

(x + 5)/(x - 2),  with the important restriction that x ≠ 4 because canceling (x - 4) is only valid when x - 4 ≠ 0.

Since x = 4 makes the original denominator zero and cancels out, it indicates a hole in the graph at x = 4.

To find the y-coordinate of the hole, substitute x = 4 into the simplified function:

y = (4 + 5)/(4 - 2) = 9/2.

Therefore, the hole in the graph is located at (4, 9/2).
There is a hole at the point (4, 9/2) on the graph of the function.

Identify any holes in the graph of the rational function: \( \frac{x^{2}+x-20}{x^{2}-6 x+8} \)