text: # Domain, zeroes, and graphing to check the function
  
  #### thebestrose828
  
  Hey,
  I really need help solving this question. If someone could provide a step-by-step guide? The question asks for the domain of the function, zeroes, and to graph it to check the function.
  The function is as follows:
  x3 +x2-2x
  _________=f(x)
  2x3-x2-6x
  
  (The above is supposed to be those two equations divided by each other equals f(x), not really that computer savvy with typing math stuff)
  
  Last edited:
  
  #### skeeter
  
  MHF Helper
  start by factoring numerator & denominator ...
  
  $f(x) = \dfrac{x(x+2)(x-1)}{x(2x+3)(x-2)}$
  
  note that the numerator & denominator have a common factor of $x$ ... this indicates a point discontinuity (a "hole") at $(0,1/3)$, also indicating the absence of a y-intercept.
  
  vertical asymptotes where $2x+3=0 \implies x = -\dfrac{3}{2}$ and where $x-2=0 \implies x = 2$
  
  so, domain is all real values of $x$ except $x=0$, $x=-\dfrac{3}{2}$, and $x=2$
  
  horizontal asymptote is determined by the ratio of the leading coefficients, $y = \dfrac{1}{2}$
  
  zeros where $x+2=0 \implies x=-2$ and $x-1=0 \implies x=1$
  
  #### Attachments
  
  • 9.6 KB Views: 3
explanation: The document contains a math question and a detailed answer provided by a user named skeeter. The question asks for the domain, zeroes, and a graph of the function \( f(x) = \frac{x^3 + x^2 - 2x}{2x^3 - x^2 - 6x} \). The answer includes factoring the function, identifying the domain, vertical and horizontal asymptotes, and the zeroes of the function.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Domain, zeroes, and graphing to check the function
  
  #### thebestrose828
  
  Hey,
  I really need help solving this question. If someone could provide a step-by-step guide? The question asks for the domain of the function, zeroes, and to graph it to check the function.
  The function is as follows:
  x3 +x2-2x
  _________=f(x)
  2x3-x2-6x
  
  (The above is supposed to be those two equations divided by each other equals f(x), not really that computer savvy with typing math stuff)
  
  Last edited:
  
  #### skeeter
  
  MHF Helper
  start by factoring numerator & denominator ...
  
  $f(x) = \dfrac{x(x+2)(x-1)}{x(2x+3)(x-2)}$
  
  note that the numerator & denominator have a common factor of $x$ ... this indicates a point discontinuity (a "hole") at $(0,1/3)$, also indicating the absence of a y-intercept.
  
  vertical asymptotes where $2x+3=0 \implies x = -\dfrac{3}{2}$ and where $x-2=0 \implies x = 2$
  
  so, domain is all real values of $x$ except $x=0$, $x=-\dfrac{3}{2}$, and $x=2$
  
  horizontal asymptote is determined by the ratio of the leading coefficients, $y = \dfrac{1}{2}$
  
  zeros where $x+2=0 \implies x=-2$ and $x-1=0 \implies x=1$
  
  #### Attachments
  
  • 9.6 KB Views: 3
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer provided by a user named skeeter. The question asks for the domain, zeroes, and a graph of the function \( f(x) = \frac{x^3 + x^2 - 2x}{2x^3 - x^2 - 6x} \). The answer includes factoring the function, identifying the domain, vertical and horizontal asymptotes, and the zeroes of the function.
  
  Label: question-and-answer