user: How could I answer something like this:
Determine whether the following function is even or odd and the type of symmetry it has.
f left parenthesis x right parenthesis equals space open curly brackets table row cell x plus 7 end cell cell negative 6 space less than space x space less than 0 end cell row cell negative x plus 7 end cell cell 0 less or equal than x less than 6 end cell end table close.
This function is Blank 1 and has Blank 2 symmetry. [Timestamp: 2024/12/02 17:58:20]
ai: To determine whether the function ( f(x) ) is even or odd, and to identify its type of symmetry, you can follow these steps:
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Definition of Even and Odd Functions:
- A function is even if ( f(-x) = f(x) ) for all ( x ) in the domain.
- A function is odd if ( f(-x) = -f(x) ) for all ( x ) in the domain.
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Evaluate ( f(-x) ):
- For ( x < 0 ): ( f(-x) = f(x + 7) ) (since ( -x > 0 )).
- For ( 0 \leq x < 6 ): ( f(-x) = -(-x + 7) = x - 7 ).
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Check the intervals:
- For ( x < 0 ): ( f(x) = x + 7 ) and ( f(-x) = -x + 7 ). Check if ( f(-x) = f(x) ) or ( f(-x) = -f(x) ).
- For ( 0 \leq x < 6 ): ( f(x) = -x + 7 ) and ( f(-x) = x + 7 ). Again, check the conditions.
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Determine Symmetry:
- If ( f(-x) = f(x) ), the function is even and has y-axis symmetry.
- If ( f(-x) = -f(x) ), the function is odd and has origin symmetry.
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Conclusion:
- After evaluating the function in both intervals, you can conclude whether it is even, odd, or neither, and identify the type of symmetry.
Based on the analysis, you can fill in the blanks:
This function is neither even nor odd and has no symmetry.
If you find that it meets the criteria for either even or odd, adjust the answer accordingly. [Timestamp: 2024/12/02 17:58:28]