QuestionClassify the functions as Odd, Even, or Neither: $f(x)=\sqrt{x}$ , $g(x)=4$ , $h(x)=\frac{4}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We are given three functions $f(x)=\sqrt{x}$ , $g(x)=4$ , and $h(x)=\frac{4}{x}$ . We need to determine if each function is odd, even, or neither

STEP 2

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$ .
A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
If a function is neither even nor odd, then it does not satisfy either of the above conditions.

STEP 3

Let's start with the function $f(x)=\sqrt{x}$ . We need to check if $f(-x)$ equals to $f(x)$ or $-f(x)$ .

STEP 4

Calculate $f(-x)$ for the function $f(x)=\sqrt{x}$ .
$f(-x) = \sqrt{-x}$

STEP 5

The square root of a negative number is not a real number. Therefore, $f(-x)$ is not defined for all $x$ in the domain of $f$ . Hence, $f(x)=\sqrt{x}$ is neither even nor odd.

STEP 6

Now let's move on to the function $g(x)=4$ . We need to check if $g(-x)$ equals to $g(x)$ or $-g(x)$ .

STEP 7

Calculate $g(-x)$ for the function $g(x)=4$ .
$g(-x) =4$

STEP 8

Since $g(-x) = g(x)$ for all $x$ in the domain of $g$ , $g(x)=4$ is an even function.

STEP 9

Finally, let's consider the function $h(x)=\frac{4}{x}$ . We need to check if $h(-x)$ equals to $h(x)$ or $-h(x)$ .

STEP 10

Calculate $h(-x)$ for the function $h(x)=\frac{4}{x}$ .
$h(-x) = \frac{4}{-x} = -\frac{4}{x} = -h(x)$

STEP 11

Since $h(-x) = -h(x)$ for all $x$ in the domain of $h$ , $h(x)=\frac{4}{x}$ is an odd function.
In conclusion, $f(x)=\sqrt{x}$ is neither even nor odd, $g(x)=4$ is even, and $h(x)=\frac{4}{x}$ is odd.

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