Determine whether the function is even, odd, or neither. Then determine whether the func

Determine whether the function is even, odd, or neither. Choose the correct answer below
odd
neither
even
Determine whether the graph of the function is symmetric with respect to the -axis, the or
origin
neither
y-axis

Ask by Hamilton Ruiz. in the United States

Mar 25,2025

Question

Determine whether the function is even, odd, or neither. Then determine whether the func

Determine whether the function is even, odd, or neither. Choose the correct answer below
odd
neither
even
Determine whether the graph of the function is symmetric with respect to the -axis, the or
origin
neither
y-axis

Ask by Hamilton Ruiz. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

1. Compute $$ h(-x) $$:
   
   $$
   h(-x) = (-x)^{6} - (-x)^{8}
   $$
   
2. Simplify each term using properties of exponents:
   
   $$
   (-x)^{6} = x^{6} \quad \text{(since $$ 6 $$ is even)}
   $$
   
   $$
   (-x)^{8} = x^{8} \quad \text{(since $$ 8 $$ is even)}
   $$
   
3. Substitute the simplified terms back into $$ h(-x) $$:

$$
   h(-x) = x^{6} - x^{8}
   $$
   
4. Compare $$ h(-x) $$ with $$ h(x) $$:

$$
   h(x) = x^{6} - x^{8}
   $$
   
   Since $$ h(-x) = h(x) $$ for all $$ x $$, the function is even.
   
5. Determine the symmetry of the graph:

Because the function is even, its graph is symmetric with respect to the $$ y $$-axis.

6. Conclusion:
   - The function is even.
   - The graph is symmetric with respect to the $$ y $$-axis.
The function $$ h(x) = x^{6} - x^{8} $$ is even and its graph is symmetric with respect to the $$ y $$-axis.

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