Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x+1}{x^{2}+x+1} $$

Short answer

The function \( f(x)=\frac{x+1}{x^{2}+x+1} \) has a horizontal asymptote at \( y = 0 \) and no vertical asymptotes or extrema.

Step-by-step solution

Find the Vertical Asymptotes

Vertical asymptotes occur at the x-values which make the denominator zero. So first, solve \( x^{2}+x+1 = 0 \). This has no solution because the discriminant \( b^{2}-4ac \) yields \( 1-4 < 0 \), hence the function has no vertical asymptote.

Find the Horizontal Asymptote

To find the horizontal asymptote, look at the degree of the polynomials in the numerator and denominator. Since the degree of the denominator (2) is greater than the degree of the numerator (1), the horizontal asymptote is \( y = 0 \).

Find the Extrema

The extrema of the function can be found by taking the derivative of the function and setting it equal to 0. However, given that our function is rational, it's extrema is mainly defined by its asymptotes and undefined points. Since we established previously that there are no defined vertical asymptotes or undefined points, the function doesn't have any relative maxima or minima within its domain.

Plotting the Function

Using a graphing utility, one can plot \( f(x) = \frac{x+1}{x^{2}+x+1} \). The graph will confirm the horizontal asymptote at \( y = 0 \) and that there are no vertical asymptotes or extrema.