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