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

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

Ask by Fleming Turnbull. in the United States

Mar 25,2025

Question

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

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

Ask by Fleming Turnbull. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

**Step 1. Compute $$ h(-x) $$:**

Given
$$
h(x) = x^6 - x^8,
$$
we substitute $$-x$$ for $$ x $$:
$$
h(-x) = (-x)^6 - (-x)^8.
$$

**Step 2. Simplify $$ h(-x) $$:**

Since $$(-x)^6 = x^6$$ (even exponent) and $$(-x)^8 = x^8$$ (even exponent), we obtain:
$$
h(-x) = x^6 - x^8.
$$

**Step 3. Compare $$ h(-x) $$ with $$ h(x) $$:**

We see that:
$$
h(-x) = x^6 - x^8 = h(x).
$$
Thus, the function satisfies the condition $$ h(-x) = h(x) $$ for all $$ x $$.

**Step 4. Conclusion on Even/Odd:**

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

**Step 5. Determine the symmetry of the graph:**

A function that is even is symmetric with respect to the $$ y $$-axis.

**Final Answers:**

- The function is **even**.
- Its 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**.

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

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

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