Questions: For the given rational function, (A) Find the intercepts for the graph. (B) Determine the domain. (C) Find any vertical or horizontal asymptotes for the graph. (D) Sketch any asymptotes as dashed lines. Then sketch a graph of y=f(x). f(x)= (x-6)/(x+6) (A) What are the x-intercepts? Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The x-intercept(s) is/are . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no x-intercepts.

For the given rational function, (A) Find the intercepts for the graph. (B) Determine the domain. (C) Find any vertical or horizontal asymptotes for the graph. (D) Sketch any asymptotes as dashed lines. Then sketch a graph of y=f(x). f(x)= (x-6)/(x+6) (A) What are the x-intercepts? Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The x-intercept(s) is/are . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no x-intercepts.
Transcript text: Part 1 of 6 Points: 0 of 1 Save For the given rational function, (A) Find the intercepts for the graph. (B) Determine the domain. (C) Find any vertical or horizontal asymptotes for the graph. (D) Sketch any asymptotes as dashed lines. Then sketch a graph of $y=f(x)$. \[ f(x)=\frac{x-6}{x+6} \] (A) What are the $x$-intercepts? Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The $x$-intercept(s) is/are $\square$ . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no $x$-intercepts.
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Solution

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Solution Steps

Step 1: Find the x-intercepts

To find the $x$-intercepts, set $f(x) = 0$ and solve for $x$: \[ \frac{x-6}{x+6} = 0 \] This implies $x - 6 = 0$, so: \[ x = 6 \]

Step 2: Determine the domain

The domain of the function is all real numbers except where the denominator is zero: \[ x + 6 \neq 0 \implies x \neq 6 \] Thus, the domain is: \[ (-\infty, -6) \cup (-6, \infty) \]

Step 3: Find vertical and horizontal asymptotes

Vertical asymptotes occur where the denominator is zero: \[ x + 6 = 0 \implies x = -6 \] Horizontal asymptotes are determined by the degrees of the numerator and denominator. Since both are linear (degree 1), the horizontal asymptote is the ratio of the leading coefficients: \[ y = \frac{1}{1} = 1 \]

Final Answer

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -2, "ymax": 2}, "commands": ["y = (x-6)/(x+6)", "y = 1"], "latex_expressions": ["$y = \\frac{x-6}{x+6}$", "$y = 1$"]}

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