QuestionClassify the functions as Odd, Even, or Neither: $f(x)=(x-3)^{2}$ , $g(x)=x^{2}+3$ , $h(x)=\frac{3}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We have three functions to evaluate $f(x)=(x-3)^{}$ , $g(x)=x^{}+3$ , and $h(x)=\frac{3}{x}$ . . 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 does not satisfy either of the above conditions.

STEP 2

First, let's evaluate the function $f(x)=(x-)^{2}$ for $f(-x)$ and compare it with $f(x)$ .
$f(-x) = ((-x)-)^{2}$

STEP 3

implify the expression for $f(-x)$ .
$f(-x) = (-(x+3))^{2}$

STEP 4

implify further.
$f(-x) = (x+3)^{2}$

STEP 5

Since $(x+3)^{2}$ is not equal to $(x-3)^{2}$ , $f(x)$ is not even.

STEP 6

Now, let's check if $f(x)$ is odd. For this, we need to compare $f(-x)$ with $-f(x)$ .
$-f(x) = -((x-3)^{2})$

STEP 7

Since $(x+3)^{2}$ is not equal to $-((x-3)^{2})$ , $f(x)$ is not odd.

STEP 8

Since $f(x)$ is neither even nor odd, we can conclude that $f(x)$ is neither.

STEP 9

Now, let's evaluate the function $g(x)=x^{2}+3$ for $g(-x)$ and compare it with $g(x)$ .
$g(-x) = (-x)^{2}+3$

STEP 10

implify the expression for $g(-x)$ .
$g(-x) = x^{2}+3$

STEP 11

Since $x^{}+3$ is equal to $x^{}+3$ , $g(x)$ is even.

STEP 12

Now, let's evaluate the function $h(x)=\frac{}{x}$ for $h(-x)$ and compare it with $h(x)$ .
$h(-x) = \frac{}{-x}$

STEP 13

implify the expression for $h(-x)$ .
$h(-x) = -\frac{3}{x}$

STEP 14

Since $-\frac{3}{x}$ is equal to $-h(x)$ , $h(x)$ is odd.
In conclusion, $f(x)$ is neither even nor odd, $g(x)$ is even, and $h(x)$ is odd.

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