Question Determine if the function $m(x) = -5x^5 + 3x^3 + x$ is even, odd, or neither.

Studdy Solution

STEP 1

Assumptions
1. A function $f(x)$ is even if $f(x) = f(-x)$ for all $x$ in the domain of $f$ .
2. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
3. If a function is neither even nor odd, then it does not satisfy either of the above conditions.
4. The function given is $m(x)=-5x^{5}+3x^{3}+x$ .

STEP 2

To determine if the function $m(x)$ is even, we will substitute $-x$ for $x$ and simplify to see if the resulting expression is equal to $m(x)$ .
$m(-x)=-5(-x)^{5}+3(-x)^{3}+(-x)$

STEP 3

Now, simplify the expression by applying the power rule and the fact that the negative sign raised to an odd power is negative.
$m(-x)=-5(-1)^{5}x^{5}+3(-1)^{3}x^{3}+(-1)x$

STEP 4

Since $(-1)^{5} = -1$ and $(-1)^{3} = -1$ , the expression simplifies to:
$m(-x)=-5(-1)x^{5}+3(-1)x^{3}-x$

STEP 5

Simplify the expression further by multiplying through by $-1$ .
$m(-x)=5x^{5}-3x^{3}-x$

STEP 6

Compare $m(-x)$ with $m(x)$ to see if they are equal.
Since $m(-x) = 5x^{5}-3x^{3}-x$ and $m(x) = -5x^{5}+3x^{3}+x$ , we can see that $m(-x) \neq m(x)$ , so the function is not even.

STEP 7

To determine if the function $m(x)$ is odd, we will compare $m(-x)$ with $-m(x)$ .
$-m(x) = -(-5x^{5}+3x^{3}+x)$

STEP 8

Distribute the negative sign through the expression.
$-m(x) = 5x^{5}-3x^{3}-x$

STEP 9

Now compare $m(-x)$ with $-m(x)$ .
Since $m(-x) = 5x^{5}-3x^{3}-x$ and $-m(x) = 5x^{5}-3x^{3}-x$ , we see that $m(-x) = -m(x)$ , so the function is odd.
The function $m(x)=-5x^{5}+3x^{3}+x$ is odd.

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