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:
-
\( f(x) = -\frac{1}{2} x^4 + 5 \)
- 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.
-
\( f(x) = -8x^3 + 5x \)
- 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.
-
\( f(x) = \frac{-4}{x^3 - x + 1} \)
- 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.
-
\( f(x) = \frac{x^5}{x^4 - 1} \)
- 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.
-
\( f(x) = -\sqrt{2x} \)
- 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.
-
\( f(x) = \sqrt[3]{x} - x^3 \)
- 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:
- \( f(x) = -8x^3 + 5x \)
- \( f(x) = \frac{x^5}{x^4 - 1} \)
- \( f(x) = \sqrt[3]{x} - x^3 \)