Chapter 2.5, Problem 44AYU

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.

$h\left(x\right)\text{ = }\sqrt{x\text{ + 1}}$

To determine

To graph: The function $f\left(x\right)=\sqrt{x+1}$, 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 44AYU

Domain of the function $f\left(x\right)=\sqrt{x+1}$ is $\left[-1,\infty \right)$.

Range of the function $f\left(x\right)=\sqrt{x+1}$ is $\left[0,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=\sqrt{x+1}$

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 $f\left(x\right)=\sqrt{x+1}$, replace $x$ by $x+1$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=\sqrt{x}$, that it is shifted left 1 unit.

$f\left(x\right)=\sqrt{x+1}$     replace $x$ by $x+1$; Horizontal shift left 1 unit.

Interpretation:

Domain of the function $f\left(x\right)=\sqrt{x+1}$ is $\left[-1,\infty \right)$.

Range of the function $f\left(x\right)=\sqrt{x+1}$ is $\left[0,\infty \right)$.