Find any asymptotes and holes in the graph of the rational function. Verify your answers by using a graphing utility. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)f(x) = x2 + 2x + 17x2 − x − 8

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To find the asymptotes and holes in the graph of the rational function, we first need to simplify the function and put it in the standard form.

The given function is f(x) = (x^2 + 2x + 1) / (x^2 - x - 8).

Step 1: Factor the numerator and the denominator.

The numerator x^2 + 2x + 1 can be factored into (x+1)^2.

The denominator x^2 - x - 8 can be factored into (x-4)(x+2).

So, f(x) = (x+1)^2 / [(x-4)(x+2)].

Step 2: Look for any common factors in the numerator and the denominator.

There are no common factors, so there are no holes in the graph.

Step 3: Find the vertical asymptotes.

The vertical asymptotes occur where the denominator equals zero. So, set the denominator equal to zero and solve for x.

(x-4)(x+2) = 0

Setting each factor equal to zero gives x = 4 and x = -2. So, the vertical asymptotes are x = 4 and x = -2.

Step 4: Find the horizontal asymptote.

The degree of the numerator is 2 and the degree of the denominator is 2, so the horizontal asymptote is the ratio of the leading coefficients, which is 1.

So, the horizontal asymptote is y = 1.

Step 5: Verify your answers using a graphing utility.

You can use a graphing calculator or an online graphing tool to plot the function and verify that the asymptotes and holes are where you calculated them to be.

So, the function f(x) = (x^2 + 2x + 1) / (x^2 - x - 8) has vertical asymptotes at x = 4 and x = -2, a horizontal asymptote at y = 1, and no holes.