Chapter 2.5, Problem 56AYU

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.

56. $h\left(x\right)\text{ = }\frac{4}{x}\text{ + 2}$

To determine

To graph: The function $f\left(x\right)= \frac{4}{x}+2$, 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 56AYU

Domain of the function $f\left(x\right)= \frac{4}{x}+2$ is $(-\infty ,0)\text{U}\left(0,\infty \right)$.

Range of the function $f\left(x\right)= \frac{4}{x}+2$ is $(-\infty ,0)\text{U}\left(0,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)= \frac{4}{x}+2$

Graph:

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

Step 1: The function $f\left(x\right)=\frac{1}{x}$ is the reciprocal function.

$f\left(x\right)=\frac{1}{x}$     reciprocal function

Step 2: To obtain the graph of $f\left(x\right)= \frac{4}{x}$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\frac{1}{x}$, by 4, that it is vertically stretched by the factor of 4.

$f\left(x\right)= \frac{4}{x}$     multiply by 4, vertically stretched by a factor of 4

Step 3: To obtain the graph of $f\left(x\right)= \frac{4}{x}+2$, add 2 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)= \frac{4}{x}$, that it is shifted up 2 units.

$f\left(x\right)= \frac{4}{x}+2$     add 2; vertically shift up 2 units.

Interpretation:

Domain of the function $f\left(x\right)= \frac{4}{x}+2$ is $(-\infty ,0)\text{U}\left(0,\infty \right)$.

Range of the function $f\left(x\right)= \frac{4}{x}+2$ is $(-\infty ,0)\text{U}\left(0,\infty \right)$.