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 can recall that a function \( f(x) \) is considered odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of the function. Let's evaluate each of the provided functions accordingly:

  1. \( f(x) = -\frac{1}{2} x^4 + 5 \)

    • Check \( f(-x) = -\frac{1}{2} (-x)^4 + 5 = -\frac{1}{2} x^4 + 5 \)
    • This is equal to \( f(x) \), so this function is not odd.
  2. \( f(x) = -8x^3 + 5x \)

    • Check \( f(-x) = -8(-x)^3 + 5(-x) = 8x^3 - 5x = -(-8x^3 + 5x) \)
    • This is equal to \(-f(x)\), so this function is odd.
  3. \( f(x) = \frac{-4}{x^3 - x + 1} \)

    • Check \( f(-x) = \frac{-4}{(-x)^3 - (-x) + 1} = \frac{-4}{-x^3 + x + 1} \)
    • To determine if this is odd, we compute \(-f(x) = \frac{4}{x^3 - x + 1}\).
    • We need \( \frac{-4}{-x^3 + x + 1} = \frac{4}{x^3 - x + 1}\) for it to hold; since this changes the signs and doesn’t match our findings, this function is not odd.
  4. \( f(x) = \frac{x^5}{x^4 - 1} \)

    • Check \( f(-x) = \frac{(-x)^5}{(-x)^4 - 1} = \frac{-x^5}{x^4 - 1} \)
    • This is equal to \(-f(x)\), so this function is odd.
  5. \( f(x) = -\sqrt{2x} \)

    • Check \( f(-x) = -\sqrt{2(-x)} \)
    • This is not defined for positive values of \( x \). While \( f(-x) \) would suggest it's odd if it were defined, we cannot consider it odd due to its undefined nature; thus, this function is not odd.
  6. \( f(x) = \sqrt[3]{x} - x^3 \)

    • Check \( f(-x) = \sqrt[3]{-x} - (-x)^3 = -\sqrt[3]{x} + x^3 \)
    • This seems to align to \( -(\sqrt[3]{x} - x^3) \) as \(-f(x)\), so this function is odd.

In Summary:

The functions that are odd are:

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

Odd Functions Selected:

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