QuestionDetermine if the function $f(x)=x^{5}-7 x$ is even, odd, or neither.

Studdy Solution

STEP 1

Assumptions1. The function is $f(x) = x^{5} -7x$ . 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 $f$

STEP 2

First, we need to find $f(-x)$ .
$f(-x) = (-x)^{5} -7(-x)$

STEP 3

implify the expression.
$f(-x) = -x^{5} +7x$

STEP 4

Now, we need to compare $f(-x)$ with $f(x)$ and $-f(x)$ to determine if the function is even, odd, or neither.

STEP 5

First, let's compare $f(-x)$ with $f(x)$ .
$f(x) = x^{5} -7x$ $f(-x) = -x^{5} +7x$ We can see that $f(-x)$ is not equal to $f(x)$ , so the function is not even.

STEP 6

Next, let's compare $f(-x)$ with $-f(x)$ .
$-f(x) = -(x^{5} -x) = -x^{5} +x$ $f(-x) = -x^{5} +x$ We can see that $f(-x)$ is equal to $-f(x)$ , so the function is odd.
The function $f(x) = x^{5} -x$ is odd.

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