Chapter 2.5, Problem 55AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

55. $h\left(x\right)\text{ = }\sqrt{-x}\text{ - 2}$

To determine

To graph: The function $f\left(x\right)= \sqrt{-x}-2$, using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

Answer to Problem 55AYU

Domain of the function $f\left(x\right)= \sqrt{-x}-2$ is $\left(-\infty ,0\right]$.

Range of the function $f\left(x\right)= \sqrt{-x}-2$ is $\left[-2,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)= \sqrt{-x}-2$

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$.

Step 1: The function $f\left(x\right)=\sqrt{x}$ is the square root function.

$f\left(x\right)=\sqrt{x}$     square root function

Step 2: To obtain the graph of $f\left(x\right)= \sqrt{-x}$, replace $x$ by $-x$ from each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\sqrt{x}$, by, that it is reflection about the $y\text{-axis}$.

$f\left(x\right)= \sqrt{-x}$     replace $x$ by $-x$, reflect about the $y\text{-axis}$

Step 3: To obtain the graph of $f\left(x\right)= \sqrt{-x}-2$, subtract 2 from each $y\text{-coordinate}$ of the graph of $f\left(x\right)= \sqrt{-x}$, that it is shifted down 2 units.

$f\left(x\right)= \sqrt{-x}-2$     subtract 2; vertically shift down 2 units

Interpretation:

Domain of the function $f\left(x\right)= \sqrt{-x}-2$ is $\left(-\infty ,0\right]$.

Range of the function $f\left(x\right)= \sqrt{-x}-2$ is $\left[-2,\infty \right)$.