Chapter 2, Problem 65RE

To determine

To sketch: the graph of the function. Also state the domain, intercepts and range of the graph.

Answer to Problem 65RE

The intercept is $\left(0,-2\right)$ , and $\left(0.3,0\right)$

The range of $F\left(x\right)$ is the set of all real numbers.

The intercepts are $\left(0,-2\right)$ , and $\left(0.3,0\right)$ , domain is the set of all real numbers is the set of all real numbers.

Explanation of Solution

Given information:

Given function

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

Calculation:

Graph the basic function $g\left(x\right)=x^3$ .

  

Shift the graph of $g\left(x\right)=x^3$ , horizontally right 1 unit to obtain ${\left(x-1\right)}^3$

  

Shift the graph of $g\left(x\right)=3{\left(x-1\right)}^3$ by multiplying each $y$ coordinate of $g\left(x\right)=3{\left(x-1\right)}^3$ by a factor 3.

  

Draw the graph $g\left(x\right)=3{\left(x-1\right)}^3+1$ by shifting the graph of $g\left(x\right)=3{\left(x-1\right)}^3$ , vertically up 1 unit.

  

Intercepts are the points where the graph touches or crosses the coordinate axes. In the graph $\left(0,-2\right)$ touches the $y$ axis and $\left(0.3,0\right)$ touches the $x$ axis. So, the intercept is $\left(0,-2\right)$ , and $\left(0.3,0\right)$

The points on the graph have $y$ coordinates between $-\infty$ and $\infty$ .So, the range of $F\left(x\right)$ is the set of all real numbers.

Therefore, the intercepts are $\left(0,-2\right)$ , and $\left(0.3,0\right)$ , domain is the set of all real numbers is the set of all real numbers.