Question

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

Answer 1

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