Math Problem Statement
Question content area top Part 1 Determine algebraically whether the given function is even, odd, or neither. f left parenthesis x right parenthesis equals RootIndex 9 StartRoot 3 x EndRootf(x)=93x Question content area bottom Part 1 Even Neither NeitherNeither Odd OddOdd
Solution
To determine whether the function is even, odd, or neither, we need to use the following definitions:
- Even Function: A function is even if for all .
- Odd Function: A function is odd if for all .
Step 1: Find
To check if the function is even or odd, we first compute :
[ f(-x) = \sqrt[9]{3(-x)} = \sqrt[9]{-3x} ]
Step 2: Compare with
-
Even Function Test: For to be even, we would need . However, , which is not equal to because the exponents of and will result in different values.
-
Odd Function Test: For to be odd, we would need . We see that , and since the 9th root of is the negative of the 9th root of , we have:
[ \sqrt[9]{-3x} = -\sqrt[9]{3x} ]
This matches , so the function is odd.
Conclusion:
The function is an odd function.
Would you like a more detailed explanation or examples of even and odd functions? Here are some related questions for you to consider:
- How do you determine if a function is even or odd graphically?
- Can a function be both even and odd? If so, provide an example.
- How would you determine if a function involving absolute values is even or odd?
- What are some real-world examples of even and odd functions?
- How do transformations affect whether a function is even or odd?
Tip: Always check the symmetry of the graph when determining if a function is even or odd. Even functions are symmetric about the y-axis, while odd functions have rotational symmetry about the origin.
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Math Problem Analysis
Mathematical Concepts
Algebra
Function Analysis
Even and Odd Functions
Formulas
f(x) = \sqrt[9]{3x}
f(-x) = \sqrt[9]{-3x}
Theorems
Even Function Test: f(-x) = f(x)
Odd Function Test: f(-x) = -f(x)
Suitable Grade Level
Grades 9-12