Questions: 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. g(x)=x^4+3x

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.

g(x)=x^4+3x

Transcript text: 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. \[ g(x)=x^{4}+3 x \]

Solution

Solution Steps

To determine if a function is even, odd, or neither, we need to evaluate the function at $-x$ and compare it to the original function $g(x)$. A function is even if $g(-x) = g(x)$ for all $x$, and it is odd if $g(-x) = -g(x)$ for all $x$. If neither condition is met, the function is neither even nor odd. For the given function $g(x) = x^4 + 3x$, we will calculate $g(-x)$ and compare it to $g(x)$ and $-g(x)$.

Step 1: Evaluate $ g(-x) $

To determine if the function $ g(x) = x^4 + 3x $ is even, odd, or neither, we first evaluate $ g(-x) $.

\[ g(-x) = (-x)^4 + 3(-x) = x^4 - 3x \]

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

Next, we compare $ g(-x) $ with $ g(x) $ and $-g(x)$.

$ g(x) = x^4 + 3x $
$-g(x) = -(x^4 + 3x) = -x^4 - 3x $

Comparing:

$ g(-x) = x^4 - 3x $ is not equal to $ g(x) = x^4 + 3x $
$ g(-x) = x^4 - 3x $ is not equal to $-g(x) = -x^4 - 3x $

Step 3: Determine the Symmetry

Since $ g(-x) \neq g(x) $ and $ g(-x) \neq -g(x) $, the function is neither even nor odd. Therefore, the graph of the function is symmetric with respect to neither the $ y $-axis nor the origin.

Final Answer

neither

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