Question

consider the following function.
r(x)=\begin{cases}-\frac{5}{8}x&\text{if }x < - 2\\frac{2}{3}sqrt3{x}&\text{if }xgeq - 2end{cases}
step 3 of 3: identify the correct graph of this piece - wise - defined function.
answer

Explanation:

Step1: Analyze the first - part of the function

For \(r(x)=-\frac{5}{8}x\) when \(x < - 2\), it is a linear function with slope \(m =-\frac{5}{8}\). When \(x=-2\), \(r(-2)=\frac{5}{4}\) (but this point is not included in this part of the function as the inequality is \(x < - 2\)).

Step2: Analyze the second - part of the function

For \(r(x)=\frac{2}{3}\sqrt[3]{x}\) when \(x\geq - 2\), the cube - root function \(\sqrt[3]{x}\) is a continuous function. When \(x = - 2\), \(r(-2)=\frac{2}{3}\sqrt[3]{-2}=-\frac{2}{3}\sqrt[3]{2}\). The cube - root function \(\sqrt[3]{x}\) has a domain of all real numbers and for \(y = \frac{2}{3}\sqrt[3]{x}\), as \(x\) increases, \(y\) also increases.

Step3: Consider the behavior at the break - point

There is a break in the function at \(x=-2\). The left - hand side function \(y =-\frac{5}{8}x\) approaches \(\frac{5}{4}\) as \(x\) approaches \(-2\) from the left, and the right - hand side function \(y=\frac{2}{3}\sqrt[3]{x}\) has the value \(-\frac{2}{3}\sqrt[3]{2}\) at \(x = - 2\).

Since no options for the graphs are provided, the general steps to identify the graph are:

1. Plot the linear part \(y =-\frac{5}{8}x\) for \(x < - 2\) as a ray with an open - circle at \(x=-2\).
2. Plot the cube - root part \(y=\frac{2}{3}\sqrt[3]{x}\) for \(x\geq - 2\) with a closed - circle at \(x=-2\).

Answer:

No specific answer can be given without the graph options. But the graph should have a ray representing \(y =-\frac{5}{8}x\) for \(x < - 2\) (open - circle at \(x=-2\)) and a curve representing \(y=\frac{2}{3}\sqrt[3]{x}\) for \(x\geq - 2\) (closed - circle at \(x=-2\)).