Math Problem Statement
Based on the ordered pairs seen in the table, make a conjecture about whether the function is even, odd, or neither. Provide clear reasoning.
Solution
The table provided shows several ordered pairs for a function :
9 | |
5 | |
-1 | |
0 | 0 |
1 | -1 |
2 | 5 |
3 | 9 |
To determine whether the function is even, odd, or neither, we need to check the symmetry properties:
- Even Function: A function is even if for all . This means that the function is symmetric with respect to the y-axis.
- Odd Function: A function is odd if for all . This means that the function is symmetric with respect to the origin.
Analysis of the Given Values:
- For , and for , . This satisfies .
- For , and for , . This also satisfies .
- For , and for , . Again, is satisfied.
- For , .
Since for every , , the function is even.
Conclusion:
The function is even because for every value of given, the function satisfies .
Would you like more details or have any questions?
Here are 5 related questions you might find interesting:
- How do you determine if a function is odd using its graph?
- What are the characteristics of a function that is neither even nor odd?
- Can a function be both even and odd? If so, under what conditions?
- How do transformations like translations affect the evenness or oddness of a function?
- How can you use calculus to verify if a function is even or odd?
Tip: To check if a function is even or odd, plug in and compare it to . If , the function is even; if , the function is odd.
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Math Problem Analysis
Mathematical Concepts
Function Analysis
Symmetry in Functions
Formulas
Even Function: f(x) = f(-x)
Odd Function: f(x) = -f(-x)
Theorems
Definition of Even and Odd Functions
Suitable Grade Level
Grades 8-10