Question

consider the following function.
q(x)=\begin{cases}\frac{1}{2x}&\text{if }x < 2\\-\frac{4}{3}x&\text{if }x>2end{cases}
step 3 of 3: identify the correct graph of this piece - wise - defined function.

Explanation:

Step1: Analyze \(q(x)=\frac{1}{2x}\) for \(x < 2\)

This is a rational - function. When \(x\) approaches \(0\) from the left, \(y\to-\infty\), and when \(x\) approaches \(0\) from the right, \(y\to+\infty\). As \(x\) approaches \(-\infty\), \(y\to0\) from the negative side, and as \(x\) approaches \(2\) from the left, \(y = \frac{1}{2\times2}=\frac{1}{4}\).

Step2: Analyze \(q(x)=-\frac{4}{3}x\) for \(x>2\)

This is a linear - function with a slope \(m =-\frac{4}{3}\). When \(x = 2\), \(y=-\frac{4}{3}\times2=-\frac{8}{3}\). The function \(y =-\frac{4}{3}x\) is a decreasing line for \(x>2\).

Step3: Consider the non - inclusion of \(x = 2\)

The function is not defined at \(x = 2\), so there are open - circles at the points corresponding to \(x = 2\) on both parts of the piece - wise function.

Since no graphs are provided, the general characteristics of the graph are: a hyperbola - like curve \(y=\frac{1}{2x}\) for \(x < 2\) (not including \(x = 2\)) and a decreasing straight - line \(y=-\frac{4}{3}x\) for \(x>2\) (not including \(x = 2\)).

Answer:

No graphs are given, but the graph has a hyperbola - like part \(y=\frac{1}{2x}\) for \(x < 2\) (open - circle at \(x = 2\)) and a decreasing linear part \(y=-\frac{4}{3}x\) for \(x>2\) (open - circle at \(x = 2\)).