In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)

Verified step by step guidance

1. Identify the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). For the function \(g(x) = \frac{2x - 4}{x + 3}\), set \(x + 3 = 0\).
2. Solve the equation \(x + 3 = 0\) to find the vertical asymptote. This will give you the value of \(x\) where the function is undefined.
3. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both the numerator \(2x - 4\) and the denominator \(x + 3\) are of degree 1, the horizontal asymptote is \(y = \frac{2}{1}\).
4. Check for a slant asymptote. Since the degrees of the numerator and denominator are equal, there is no slant asymptote for this function.
5. Graph the function using the asymptotes as guides. Plot the vertical asymptote as a dashed line at the value of \(x\) found in step 2, and the horizontal asymptote as a dashed line at \(y = 2\). Then sketch the curve of the function, approaching these asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, leading to undefined values. To find vertical asymptotes, set the denominator of the function to zero and solve for x. For the function g(x) = (2x - 4)/(x + 3), the vertical asymptote is at x = -3, where the denominator becomes zero.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote can be determined by comparing the degrees of the numerator and denominator. In g(x), since the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients, which is y = 2/1 = 2.

Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find a slant asymptote, perform polynomial long division. In the case of g(x), since the degree of the numerator (1) is not greater than that of the denominator (1), there is no slant asymptote.