text: # Even and Odd Functions
  
  They are special types of functions
  
  ## Even Functions
  
  A function is "even" when:
  
  f(x) = f(−x) for all x
  
  In other words there is symmetry about the y-axis (like a reflection):
  
  This is the curve f(x) = x2+1
  
  They got called "even" functions because the functions x2, x4, x6, x8, etc behave like that, but there are other functions that behave like that too, such as cos(x):
  
  Cosine function: f(x) = cos(x)
  It is an even function
  
  But an even exponent does not always make an even function, for example (x+1)2 is not an even function.
  
  ## Odd Functions
  
  A function is "odd" when:
  
  −f(x) = f(−x) for all x
  
  Note the minus in front of f(x): −f(x).
  
  And we get origin symmetry:
  
  This is the curve f(x) = x3−x
  
  They got called "odd" because the functions x, x3, x5, x7, etc behave like that, but there are other functions that behave like that, too, such as sin(x):
  
  Sine function: f(x) = sin(x)
  It is an odd function
  
  But an odd exponent does not always make an odd function, for example x3+1 is not an odd function.
  
  ## Neither Odd nor Even
  
  Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not have to be even or odd.
  
  In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:
  
  This is the curve f(x) = x3−x+1
  
  It is not an odd function, and it is not an even function either.
  It is neither odd nor even
  
  ## Even or Odd?
  
  ### Example: is f(x) = x/(x2−1) Even or Odd or neither?
  
  Let's see what happens when we substitute −x:
  
  f(−x) = (−x)/((−x)2−1)
  =−x/(x2−1)
  =−f(x)
  
  So f(−x) = −f(x) , which makes it an Odd Function
  
  ## Even and Odd
  
  The only function that is even and odd is f(x) = 0
  
  ## Special Properties
  
  • The sum of two even functions is even
  • The sum of two odd functions is odd
  • The sum of an even and odd function is neither even nor odd (unless one function is zero).
  
  Multiplying:
  
  • The product of two even functions is an even function.
  • The product of two odd functions is an even function.
  • The product of an even function and an odd function is an odd function.
explanation: The document provides an explanation of even and odd functions, including definitions, examples, and special properties. It includes a specific example to determine if a function is even, odd, or neither. However, it does not contain a question or an answer to a question.
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distilabel_metadata.content: # Even and Odd Functions
  
  They are special types of functions
  
  ## Even Functions
  
  A function is "even" when:
  
  f(x) = f(−x) for all x
  
  In other words there is symmetry about the y-axis (like a reflection):
  
  This is the curve f(x) = x2+1
  
  They got called "even" functions because the functions x2, x4, x6, x8, etc behave like that, but there are other functions that behave like that too, such as cos(x):
  
  Cosine function: f(x) = cos(x)
  It is an even function
  
  But an even exponent does not always make an even function, for example (x+1)2 is not an even function.
  
  ## Odd Functions
  
  A function is "odd" when:
  
  −f(x) = f(−x) for all x
  
  Note the minus in front of f(x): −f(x).
  
  And we get origin symmetry:
  
  This is the curve f(x) = x3−x
  
  They got called "odd" because the functions x, x3, x5, x7, etc behave like that, but there are other functions that behave like that, too, such as sin(x):
  
  Sine function: f(x) = sin(x)
  It is an odd function
  
  But an odd exponent does not always make an odd function, for example x3+1 is not an odd function.
  
  ## Neither Odd nor Even
  
  Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not have to be even or odd.
  
  In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:
  
  This is the curve f(x) = x3−x+1
  
  It is not an odd function, and it is not an even function either.
  It is neither odd nor even
  
  ## Even or Odd?
  
  ### Example: is f(x) = x/(x2−1) Even or Odd or neither?
  
  Let's see what happens when we substitute −x:
  
  f(−x) = (−x)/((−x)2−1)
  =−x/(x2−1)
  =−f(x)
  
  So f(−x) = −f(x) , which makes it an Odd Function
  
  ## Even and Odd
  
  The only function that is even and odd is f(x) = 0
  
  ## Special Properties
  
  • The sum of two even functions is even
  • The sum of two odd functions is odd
  • The sum of an even and odd function is neither even nor odd (unless one function is zero).
  
  Multiplying:
  
  • The product of two even functions is an even function.
  • The product of two odd functions is an even function.
  • The product of an even function and an odd function is an odd function.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides an explanation of even and odd functions, including definitions, examples, and special properties. It includes a specific example to determine if a function is even, odd, or neither. However, it does not contain a question or an answer to a question.
  
  Label: no-question