Determine whether the function is even, odd, or neither. Then determine whether the function’s graph is symmetric with resp

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

Ask by Colon Cox. in the United States

Mar 24,2025

Question

Determine whether the function is even, odd, or neither. Then determine whether the function’s graph is symmetric with resp

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

Ask by Colon Cox. in the United States

Mar 24,2025

UpStudy AI · Accepted Answer

**Step 1. Test for Even/Odd**

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

For the given function  
$$
h(x)=x^2-x^{10},
$$
compute  
$$
h(-x)=(-x)^2-(-x)^{10}.
$$

Recall that  
$$
(-x)^2=x^2,
$$
and since $$10$$ is even,  
$$
(-x)^{10}=x^{10}.
$$

So,  
$$
h(-x)=x^2-x^{10}=h(x).
$$

Since $$h(-x)=h(x)$$, the function is even.

---

**Step 2. Determine Graph Symmetry**

Even functions have graphs that are symmetric with respect to the $$y$$-axis. Therefore, the graph of the function is symmetric with respect to the $$y$$-axis.

---

**Final Answers:**

- The function is **even**.
- The graph of the function is symmetric with respect to the **$$y$$-axis**.
The function is even and its graph is symmetric with respect to the $$ y $$-axis.

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