QuestionDetermine if the graphs of $f(x)=-\sqrt{x}$ and $g(x)=\sqrt{-x}$ are identical.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)=-\sqrt{x}$ is defined for all $x \geq0$ . The function $g(x)=\sqrt{-x}$ is defined for all $x \leq0$
3. We are asked to compare the graphs of $f(x)$ and $g(x)$

STEP 2

To compare the graphs of $f(x)$ and $g(x)$ , we need to understand what each function represents.The function $f(x)=-\sqrt{x}$ represents the negative square root of $x$ . This means that for any given $x \geq0$ , $f(x)$ will be a negative number or zero.
The function $g(x)=\sqrt{-x}$ represents the square root of negative $x$ . This means that for any given $x \leq0$ , $g(x)$ will be a positive number or zero.

STEP 3

Now, let's consider the domain of each function.The domain of $f(x)$ is all $x \geq0$ . This means that $f(x)$ is defined for all non-negative real numbers.
The domain of $g(x)$ is all $x \leq0$ . This means that $g(x)$ is defined for all non-positive real numbers.

STEP 4

Next, let's consider the range of each function.The range of $f(x)$ is all $y \leq0$ . This means that $f(x)$ can take any non-positive real number as a value.
The range of $g(x)$ is all $y \geq0$ . This means that $g(x)$ can take any non-negative real number as a value.

STEP 5

Given the domains and ranges of $f(x)$ and $g(x)$ , we can see that they are mirror images of each other with respect to the x-axis.This is because $f(x)$ is defined for all $x \geq0$ and takes non-positive values, while $g(x)$ is defined for all $x \leq0$ and takes non-negative values.

STEP 6

Therefore, the graphs of $f(x)$ and $g(x)$ are not identical. They are mirror images of each other with respect to the x-axis.

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