- Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the $$ y $$-axis, the origin, or neither. $$ g(x)=x^{4}-3 x $$ Determine whether the function is even, odd, or neither. Choose the correct answer below. odd neither even

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UpStudy AI · Accepted Answer

To determine whether the function $$ g(x) = x^4 - 3x $$ is even, odd, or neither, we'll analyze its symmetry properties.

1. **Check for Evenness:**
   - A function is **even** if $$ g(-x) = g(x) $$ for all $$ x $$.
   - Calculate $$ g(-x) $$:
     $$
     g(-x) = (-x)^4 - 3(-x) = x^4 + 3x
     $$
   - Compare $$ g(-x) $$ with $$ g(x) $$:
     $$
     g(-x) = x^4 + 3x \quad \text{versus} \quad g(x) = x^4 - 3x
     $$
   - Since $$ g(-x) \neq g(x) $$, the function is **not even**.

2. **Check for Oddness:**
   - A function is **odd** if $$ g(-x) = -g(x) $$ for all $$ x $$.
   - Calculate $$ -g(x) $$:
     $$
     -g(x) = -(x^4 - 3x) = -x^4 + 3x
     $$
   - Compare $$ g(-x) $$ with $$ -g(x) $$:
     $$
     g(-x) = x^4 + 3x \quad \text{versus} \quad -g(x) = -x^4 + 3x
     $$
   - Since $$ g(-x) \neq -g(x) $$, the function is **not odd**.

3. **Conclusion on Symmetry:**
   - Since the function is neither even nor odd, its graph does not exhibit symmetry with respect to the $$ y $$-axis or the origin.

**Answer:**  
**Neither**
The function $$ g(x) = x^4 - 3x $$ is **neither even nor odd**, and its graph has **no symmetry** with respect to the $$ y $$-axis or the origin.

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