Questions: List the domain and the x-and y-intercepts of the following function. Graph the function. Be sure to label all the asymptotes. f(x) = x / (x^2 + 2x) What is the domain of the function? (-∞,-2) ∪ (-2,0) ∪ (0, ∞) (Type your answer in interval notation.) Select the correct choice below and fill in any answer boxes within your choice. A. The x-intercept(s) is/are . (Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. There is no x-intercept. Select the correct choice below and fill in any answer boxes within your choice. A. The y-intercept is (Type an ordered pair, using integers or fractions.) B. There is no y-intercept. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) C. The function has no vertical asymptotes.

List the domain and the x-and y-intercepts of the following function. Graph the function. Be sure to label all the asymptotes. f(x) = x / (x^2 + 2x) What is the domain of the function? (-∞,-2) ∪ (-2,0) ∪ (0, ∞) (Type your answer in interval notation.) Select the correct choice below and fill in any answer boxes within your choice. A. The x-intercept(s) is/are . (Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. There is no x-intercept. Select the correct choice below and fill in any answer boxes within your choice. A. The y-intercept is (Type an ordered pair, using integers or fractions.) B. There is no y-intercept. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) C. The function has no vertical asymptotes.
Transcript text: List the domain and the $x$-and $y$-intercepts of the following function. Graph the function. Be sure to label all the asymptotes. \[ f(x)=\frac{x}{x^{2}+2 x} \] What is the domain of the function? $(-\infty,-2) \cup(-2,0) \cup(0, \infty)$ (Type your answer in interval notation.) Select the correct choice below and fill in any answer boxes within your choice. A. The $x$-intercept(s) is/are $\square$. $\square$ (Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. There is no $x$-intercept. Select the correct choice below and fill in any answer boxes within your choice. A. The $y$-intercept is $\square$ (Type an ordered pair, using integers or fractions.) B. There is no $y$-intercept. Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, $\square$ $\square$ (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $\square$ and the rightmost asymptote is $\square$ (Type equations.) C. The function has no vertical asymptotes.
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Solution

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Solution Steps

Step 1: Determine the Domain

The function \( f(x) = \frac{x}{x^2 + 2x} \) has a denominator of \( x^2 + 2x \). To find the domain, we need to determine where the denominator is not equal to zero: \[ x^2 + 2x = 0 \implies x(x + 2) = 0 \implies x = 0 \text{ or } x = -2 \] Thus, the domain is: \[ (-\infty, -2) \cup (-2, 0) \cup (0, \infty) \]

Step 2: Find the x-intercept

To find the x-intercept, set \( f(x) = 0 \): \[ \frac{x}{x^2 + 2x} = 0 \implies x = 0 \] However, \( x = 0 \) is not in the domain. Therefore, there is no x-intercept.

Step 3: Find the y-intercept

To find the y-intercept, set \( x = 0 \): \[ f(0) = \frac{0}{0^2 + 2 \cdot 0} = \frac{0}{0} \] Since \( x = 0 \) is not in the domain, there is no y-intercept.

Step 4: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero: \[ x^2 + 2x = 0 \implies x = 0 \text{ or } x = -2 \] Thus, the function has vertical asymptotes at \( x = 0 \) and \( x = -2 \).

Final Answer

  • Domain: \( (-\infty, -2) \cup (-2, 0) \cup (0, \infty) \)
  • x-intercept: There is no x-intercept.
  • y-intercept: There is no y-intercept.
  • Vertical asymptotes: \( x = 0 \) and \( x = -2 \)

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 5, "ymin": -5, "ymax": 5}, "commands": ["y = x/(x**2 + 2*x)"], "latex_expressions": ["$f(x) = \\frac{x}{x^2 + 2x}$"]}

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