Chapter 2.5, Problem 57AYU

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.

57. $f\left(x\right)\text{ = -}{\left(x\text{ + 1}\right)}^3\text{ - 1}$

To determine

To graph: The function $f\left(x\right)=-{\left(x+1\right)}^3-1$, 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 57AYU

Domain of the function $f\left(x\right)=-{\left(x+1\right)}^3-1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=-{\left(x+1\right)}^3-1$ is $\left(-\infty ,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=-{\left(x+1\right)}^3-1$

Graph:

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

Step 1: The function $f\left(x\right)=x^3$ is the cube function.

$f\left(x\right)=x^3$     cube function

Step 2: To obtain the graph of $f\left(x\right)=-x^3$, multiply by $-1$ in the graph of $f\left(x\right)=x^3$, that it is reflection about the $x\text{-axis}$.

$f\left(x\right)=-x^3$     multiply by $-1$, reflection about $x\text{-axis}$

Step 3: To obtain the graph of $f\left(x\right)=-{\left(x+1\right)}^3$, replace $x$ by $x+1$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=-x^3$, that it is shifted left 1 unit.

$f\left(x\right)=-{\left(x+1\right)}^3$     replace $x$ by $x+1$; horizontal shift left 1 unit.

Step 4: To obtain the graph of $f\left(x\right)=-{\left(x+1\right)}^3-1$, subtract 1 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=-{\left(x+1\right)}^3$, that it is shifted down 1 unit.

$f\left(x\right)=-{\left(x+1\right)}^3-1$     subtract 1, vertically shift down 1 unit.

Interpretation:

Domain of the function $f\left(x\right)=-{\left(x+1\right)}^3-1$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=-{\left(x+1\right)}^3-1$ is $\left(-\infty ,\infty \right)$.