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
To determine whether the function is even, odd, or neither, we can use the following properties:
- Even Function: for all in the domain.
- Odd Function: for all in the domain.
- Neither: If the function does not satisfy either of the above properties.
Let's analyze the given table of values:
x & f(x) \\ \hline -3 & 9 \\ -2 & 4 \\ -1 & 0 \\ 0 & 0 \\ 1 & 4 \\ 2 & -5 \\ 3 & -9 \\ \end{array}$$ ### Step-by-Step Analysis: 1. **Check for Even Function:** - Compare $$f(x)$$ with $$f(-x)$$ for corresponding values of $$x$$. - For $$x = -3$$, $$f(-3) = 9$$ and $$f(3) = -9$$. Since $$f(3) \neq f(-3)$$, the function is not even. 2. **Check for Odd Function:** - Check if $$f(-x) = -f(x)$$ for corresponding values of $$x$$. - For $$x = -3$$, $$f(-3) = 9$$ and $$-f(3) = 9$$. This checks out. - For $$x = -2$$, $$f(-2) = 4$$ and $$-f(2) = 5$$. This does not satisfy the odd function condition since $$4 \neq 5$$. Since the function does not satisfy the conditions for being either even or odd, the function is **neither even nor odd**. Would you like more details on this analysis, or do you have any questions? Here are some related questions you might consider: 1. How would the table need to be modified for the function to be even? 2. What changes in the function's values would make it odd? 3. Can a function be both even and odd? Why or why not? 4. How would you represent this function graphically, and what would it tell you? 5. How does symmetry relate to even and odd functions? **Tip:** A function that is neither even nor odd often lacks symmetry, which can be confirmed by analyzing the graph or the values of the function as we did here.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Functions
Even and Odd Functions
Formulas
f(x) = f(-x) (Even function)
f(-x) = -f(x) (Odd function)
Theorems
Properties of Even and Odd Functions
Suitable Grade Level
Grades 9-12