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. \( f(x)=4 x^{2}+x^{4}+3 \) Determine whether the function is even, odd, or neither. Choose the correct answer below. even neither odd

To determine whether the function \( f(x) = 4x^{2} + x^{4} + 3 \) is even, odd, or neither, we'll use the definitions of even and odd functions: - **Even Function:** A function \( f \) is even if \( f(-x) = f(x) \) for all \( x \). - **Odd Function:** A function \( f \) is odd if \( f(-x) = -f(x) \) for all \( x \). Let's evaluate \( f(-x) \): \[ f(-x) = 4(-x)^{2} + (-x)^{4} + 3 = 4x^{2} + x^{4} + 3 = f(x) \] Since \( f(-x) = f(x) \) for all \( x \), the function \( f(x) \) is **even**. **Symmetry of the Graph:** - **Even Function:** The graph of an even function is symmetric with respect to the **\( y \)-axis**. - **Odd Function:** The graph of an odd function is symmetric with respect to the **origin**. Given that \( f(x) \) is even, its graph is symmetric with respect to the **\( y \)-axis**. **Conclusion:** - **Function Type:** Even - **Graph Symmetry:** Symmetric with respect to the \( y \)-axis **Answer:** even