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 f(x)=3x9f(x) = \sqrt[9]{3x} is even, odd, or neither, we need to use the following definitions:

  1. Even Function: A function f(x)f(x) is even if f(x)=f(x)f(-x) = f(x) for all xx.
  2. Odd Function: A function f(x)f(x) is odd if f(x)=f(x)f(-x) = -f(x) for all xx.

Step 1: Find f(x)f(-x)

To check if the function is even or odd, we first compute f(x)f(-x):

[ f(-x) = \sqrt[9]{3(-x)} = \sqrt[9]{-3x} ]

Step 2: Compare f(x)f(-x) with f(x)f(x)

  • Even Function Test: For f(x)f(x) to be even, we would need f(x)=f(x)f(-x) = f(x). However, f(x)=3x9f(-x) = \sqrt[9]{-3x}, which is not equal to f(x)=3x9f(x) = \sqrt[9]{3x} because the exponents of 3x3x and 3x-3x will result in different values.

  • Odd Function Test: For f(x)f(x) to be odd, we would need f(x)=f(x)f(-x) = -f(x). We see that f(x)=3x9f(-x) = \sqrt[9]{-3x}, and since the 9th root of 3x-3x is the negative of the 9th root of 3x3x, we have:

    [ \sqrt[9]{-3x} = -\sqrt[9]{3x} ]

This matches f(x)-f(x), so the function is odd.

Conclusion:

The function f(x)=3x9f(x) = \sqrt[9]{3x} 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:

  1. How do you determine if a function is even or odd graphically?
  2. Can a function be both even and odd? If so, provide an example.
  3. How would you determine if a function involving absolute values is even or odd?
  4. What are some real-world examples of even and odd functions?
  5. 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