Question Consider the following rational function,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{x^2+x-2}{x^2-1}}
- a. What is the domain of f?
- b. What are the x and y-intercepts of f?
- c. What are the vertical and horizontal asymptotes of f, if any? Does f have any holes?
- d. Graph f(x). Make sure to include the information you found above.
| Foundations:
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| 1) What points are not in the domain of f(x)?
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| 2) How do you find the intercepts?
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| 3) How do you find the asymptotes and zeros?
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| 4) How do you determine if f has any holes?
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| Answer:
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| 1) The point that are not in the domain of f(x) are zeros of the denominator.
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| 2) To find the x-intercepts set y = 0 and solve for x. For y-intercepts set x = 0 and simplify.
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| 3) For zeros, find the zeros of the numerator. Vertical asymptotes correspond to zeros of the denominator. Horizontal asymptotes correspond to taking the limit as x goes to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty}
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| 4) Holes occur when a single value of x is a zero of both the numerator and denominator.
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Solution:
| Step 1:
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| We start by finding the zeros of the denominator since this will give us information about vertical asymptotes and the domain. The zeros of the denominator are x = -1, 1. This tells us the domain is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1) \cup (-1, 1) \cup (1, \infty)}
and the potential vertical asymptotes are x = -1 and x = 1.
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| Step 2:
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| Now we should find the y-intercepts(zeros) to determine if f has any holes, which are zeros of numerator and denominator. Thus, y-intercepts correspond to zeros of the numerator x = -2, 1. Now we know we have a hole at x = 1, and a y-intercept at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-2, 0).}
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| Step 3:
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| For the horizontal asymptote take the limit as x goes to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty.}
This tells us that the horizontal asymptote is y = 1.
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| Final Answer:
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| Domain: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1)\cup(-1, 1)\cup (1, \infty)}
The x-intercept is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-2, 0)}
and y-intercept is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,2)}
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| f does have a hole, with vertical asymptote at x = -1, and horizontal asymptote y = 1.
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