Questions: Determine if the following function is even, odd, or neither. f(x)=5 x^4+3 x^2 (a) f(-x)= (b) Thus f(x) is Select an answer and is symmetric to the Select an answer

Determine if the following function is even, odd, or neither.
f(x)=5 x^4+3 x^2
(a) f(-x)=
(b) Thus f(x) is Select an answer and is symmetric to the Select an answer

Transcript text: Determine if the following function is even, odd, or neither. \[ f(x)=5 x^{4}+3 x^{2} \] (a) $f(-x)=$ $\square$ (b) Thus $f(x)$ is Select an answer $\vee$ and is symmetric to the Select an answer $\checkmark$

Solution

Solution Steps

To determine if a function is even, odd, or neither, we need to evaluate $ f(-x) $ and compare it to $ f(x) $ and $-f(x)$. A function is even if $ f(-x) = f(x) $, odd if $ f(-x) = -f(x) $, and neither if it satisfies neither condition.

Step 1: Evaluate $ f(-x) $

Given the function $ f(x) = 5x^4 + 3x^2 $, we substitute $-x$ for $x$ to find $ f(-x) $: \[ f(-x) = 5(-x)^4 + 3(-x)^2 = 5x^4 + 3x^2 \]

Step 2: Compare $ f(-x) $ with $ f(x) $ and $-f(x)$

From the evaluation, we have: \[ f(-x) = 5x^4 + 3x^2 = f(x) \] Since $ f(-x) = f(x) $, the function is even.