Questions: Follow the seven step strategy to graph the following rational function. f(x) = -x/(x+1) (Type an equation. Use a comma to separate answers as needed.) B. There are no vertical asymptotes. Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation(s) of the horizontal asymptote(s) is/are y = -1. (Type an equation. Use a comma to separate answers as needed.) B. There is no horizontal asymptote. Plot points between and beyond each x-intercept and vertical asymptote. Find the value of the function at the given value of x. f(x) = -x/(x+1) -3, -2, -1/2, 1, 5 f(x) = -x/(x+1) (Simplify your answers.)

Solution Steps

The given function is: \[ f(x) = \frac{-x}{x+1} \]

Vertical asymptotes occur where the denominator is zero. For \( f(x) = \frac{-x}{x+1} \), the denominator \( x+1 = 0 \) when \( x = -1 \). Therefore, there is a vertical asymptote at \( x = -1 \).

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Both the numerator and the denominator are of degree 1. The horizontal asymptote is given by the ratio of the leading coefficients: \[ y = \frac{-1}{1} = -1 \]

We need to find the value of the function at \( x = -3, -2, -\frac{1}{2}, 1, 5 \).

For \( x = -3 \): \[ f(-3) = \frac{-(-3)}{-3+1} = \frac{3}{-2} = -1.5 \]

For \( x = -2 \): \[ f(-2) = \frac{-(-2)}{-2+1} = \frac{2}{-1} = -2 \]

For \( x = -\frac{1}{2} \): \[ f\left(-\frac{1}{2}\right) = \frac{-\left(-\frac{1}{2}\right)}{-\frac{1}{2} + 1} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]

For \( x = 1 \): \[ f(1) = \frac{-(1)}{1+1} = \frac{-1}{2} = -0.5 \]

For \( x = 5 \): \[ f(5) = \frac{-(5)}{5+1} = \frac{-5}{6} \approx -0.8333 \]

Final Answer