Chapter 2.5, Problem 45AYU

To determine

To graph: The function $f\left(x\right)={\left(x-1\right)}^3+2$, 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 45AYU

Domain of the function $f\left(x\right)={\left(x-1\right)}^3+2$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)={\left(x-1\right)}^3+2$ is $\left(-\infty ,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)={\left(x-1\right)}^3+2$

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

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

Step 3: To obtain the graph of $f\left(x\right)={\left(x-1\right)}^3+2$, add 2 from each $y\text{-coordinate}$ on the graph of $f\left(x\right)={\left(x-1\right)}^3$, that it is shifted up 2 units.

$f\left(x\right)={\left(x-1\right)}^3+2$     Add 2, vertically shift up 2 units.

Interpretation:

Domain of the function $f\left(x\right)={\left(x-1\right)}^3+2$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)={\left(x-1\right)}^3+2$ is $\left(-\infty ,\infty \right)$.