QuestionDetermine if the function $g(x)=-9 x^{3}+8$ is even, odd, or neither.

Studdy Solution

STEP 1

Assumptions1. We are given the function $g(x)=-9x^{3}+8$ . . A function is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$ .
3. A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
4. If a function is neither even nor odd, then it doesn't satisfy either of the above conditions.

STEP 2

First, we need to find the value of $g(-x)$ .
$g(-x) = -9(-x)^{}+8$

STEP 3

implify the expression $g(-x)$ .
$g(-x) = -9(-x^{3})+8$

STEP 4

Further simplify the expression $g(-x)$ .
$g(-x) =9x^{3}+8$

STEP 5

Now, we compare $g(-x)$ with $g(x)$ and $-g(x)$ .
The function $g(x)$ is given as $-9x^{3}+8$ .
The negative of the function $-g(x)$ is $9x^{3}-8$ .

STEP 6

We see that $g(-x)$ is not equal to $g(x)$ , so the function is not even.

STEP 7

We also see that $g(-x)$ is not equal to $-g(x)$ , so the function is not odd.
Therefore, the function $g(x)=-9x^{3}+$ is neither even nor odd.

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