Pregunta
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Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of .

Find the domain of . Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The domain is all real , except .
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
Find the -intercepts of . Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The -intercept(s) is/are at .
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B. There are no -intercepts.
Find the -intercepts of . Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The -intercept(s) is/are at
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B. There are no -intercepts.

Ask by Bernard Weston. in the United States
Mar 17,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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Summary of :
  • Domain: All real numbers except .
  • -Intercept: .
  • -Intercept: .
  • Vertical Asymptote: .
  • Horizontal Asymptote: .
Graph:
  • Crosses the -axis at .
  • Crosses the -axis at .
  • Has a vertical asymptote at and a horizontal asymptote at .

Solución

Step 1. Factor the function
The given function is
Factor the numerator and denominator:
  • Numerator:
  • Denominator:
Thus,
This simplifies to
Step 2. Find the Domain
The original denominator equals zero when
Thus, the domain of is all real numbers except .
Domain Answer:
A. The domain is all real , except .
Step 3. Find the -intercept(s)
The -intercepts occur when . From the simplified form,
the fraction is zero when the numerator equals zero (and the denominator is not zero). Set:
Since , this is valid.
-Intercept Answer:
A. The -intercept(s) is/are at .
Step 4. Find the -intercept(s)
The -intercept occurs when . Substitute :
-Intercept Answer:
A. The -intercept(s) is/are at .
Additional Graphing Information
  • Vertical Asymptote:
    The function is undefined when (because the denominator is zero), so there is a vertical asymptote at .
  • Horizontal Asymptote:
    Since the simplified function is a rational function of degree 1 in both numerator and denominator, the horizontal asymptote is the ratio of the leading coefficients:
  • -Intercept:
    The graph crosses the -axis at .
  • -Intercept:
    The graph crosses the -axis at .
These key elements summarize the graphing strategy for .

Respondido por UpStudy AI y revisado por un tutor profesional

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The Deep Dive

To analyze the function , we first simplify it. The numerator factors to , and the denominator factors to (or ). This gives us for .
Domain: The only restriction comes from the denominator. Setting gives . Thus, the domain of is all real except :
A. The domain is all real , except .
X-intercepts: To find the -intercepts, set by setting the numerator to zero:
gives and . However, cannot be an -intercept because it is not in the domain. Therefore, the only -intercept is:
A. The -intercept(s) is/are at .
Y-intercepts: To find the -intercept, substitute into the function:
. Thus, the -intercept is:
A. The -intercept(s) is/are at .

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