3 of 93 of 9 Items

Question
Which functions are odd? Select all that apply.
(2 points)
Responses

f left-parenthesis x right-parenthesis equals negative one-half x superscript 4 baseline plus 5
Image with alt text: f left-parenthesis x right-parenthesis equals negative one-half x superscript 4 baseline plus 5

f left-parenthesis x right-parenthesis equals negative 8 x cubed plus 5 x
Image with alt text: f left-parenthesis x right-parenthesis equals negative 8 x cubed plus 5 x

f left-parenthesis x right-parenthesis equals Start Fraction negative 4 over x cubed minus x plus 1 End Fraction
Image with alt text: f left-parenthesis x right-parenthesis equals Start Fraction negative 4 over x cubed minus x plus 1 End Fraction

f left-parenthesis x right-parenthesis equals Start Fraction x superscript 5 baseline over x superscript 4 baseline minus 1 End Fraction
Image with alt text: f left-parenthesis x right-parenthesis equals Start Fraction x superscript 5 baseline over x superscript 4 baseline minus 1 End Fraction

f left-parenthesis x right-parenthesis equals negative Start Root 2 x End Root
Image with alt text: f left-parenthesis x right-parenthesis equals negative Start Root 2 x End Root

f left-parenthesis x right-parenthesis equals Root Index 3 Start Root x End Root minus x cubed

1 answer

To determine which functions are odd, we need to check if they satisfy the property \( f(-x) = -f(x) \). This means that if we plug in \(-x\) into the function and it equals the negative of the original function, then the function is classified as odd.

Let's evaluate each function one by one:

  1. \( f(x) = -\frac{1}{2}x^4 + 5 \)
    Check \( f(-x) \): \[ f(-x) = -\frac{1}{2}(-x)^4 + 5 = -\frac{1}{2}x^4 + 5 \] Since \( f(-x) \neq -f(x) \), this function is not odd.

  2. \( f(x) = -8x^3 + 5x \)
    Check \( f(-x) \): \[ f(-x) = -8(-x)^3 + 5(-x) = 8x^3 - 5x \] And \( -f(x) = -(-8x^3 + 5x) = 8x^3 - 5x \), so \( f(-x) = -f(x) \). This function is odd.

  3. \( f(x) = \frac{-4}{x^3 - x + 1} \)
    Check \( f(-x) \): \[ f(-x) = \frac{-4}{(-x)^3 - (-x) + 1} = \frac{-4}{-x^3 + x + 1} \] This does not simplify to \( -f(x) \). Therefore, this function is not odd.

  4. \( f(x) = \frac{x^5}{x^4 - 1} \)
    Check \( f(-x) \): \[ f(-x) = \frac{(-x)^5}{(-x)^4 - 1} = \frac{-x^5}{x^4 - 1} \] And \( -f(x) = -\frac{x^5}{x^4 - 1} \). Therefore, \( f(-x) = -f(x) \). This function is odd.

  5. \( f(x) = -\sqrt{2x} \)
    Check \( f(-x) \): \[ f(-x) = -\sqrt{2(-x)} \] This is not defined for \( x > 0 \). Hence, \( f(-x) \) does not equal to \( -f(x) \). This function is not odd.

  6. \( f(x) = \sqrt[3]{x} - x^3 \)
    Check \( f(-x) \): \[ f(-x) = \sqrt[3]{-x} - (-x)^3 = -\sqrt[3]{x} + x^3 \] And \( -f(x) = -(\sqrt[3]{x} - x^3) = -\sqrt[3]{x} + x^3 \). So, \( f(-x) = -f(x) \). This function is odd.

To summarize, the functions that are odd:

  • \( f(x) = -8x^3 + 5x \)
  • \( f(x) = \frac{x^5}{x^4 - 1} \)
  • \( f(x) = \sqrt[3]{x} - x^3 \)

So, the final answers for which functions are odd are:

Responses to select:

  • \( f(x) = -8x^3 + 5x \)
  • \( f(x) = \frac{x^5}{x^4 - 1} \)
  • \( f(x) = \sqrt[3]{x} - x^3 \)
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