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

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 ori
origin
neither
y-axis

Question

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

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 ori
origin
neither
y-axis

UpStudy AI · Accepted Answer

**Step 1. Check if the function is even or odd.**

The function provided is  
$$
h(x)=x^{6}-x^{8}.
$$

A function is even if  
$$
h(-x)=h(x)
$$  
and odd if  
$$
h(-x)=-h(x).
$$

Compute $$ h(-x) $$:  
$$
h(-x)=(-x)^{6}-(-x)^{8}.
$$

Recall that an even power makes the sign positive:  
$$
(-x)^6=x^6 \quad \text{and} \quad (-x)^8=x^8.
$$

Thus,  
$$
h(-x)=x^{6}-x^{8}.
$$

We see that  
$$
h(-x)=h(x),
$$  
which confirms that the function is even.

**Step 2. Determine the symmetry of the graph.**

For even functions, the graph is symmetric with respect to the $$ y $$-axis.

**Final Answers:**

- 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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