Question

consider the following function.
q ( x ) = left{ \begin{array} { l l } { - \frac { 1 } { 5 } x } & { \text { if } x < - 3 } \\ { \frac { 5 } { 3 } sqrt 5 { x } } & { \text { if } x geq - 3 } end{array}
ight.
step 3 of 3: identify the correct graph of this piecewise - defined function.
answer
keypad
keyboard shortcuts

Explanation:

Step1: Analyze the first part of the piece - wise function

For \(y =-\frac{1}{5}x\) when \(x < - 3\). When \(x=-3\), \(y =-\frac{1}{5}\times(-3)=\frac{3}{5}\). Since \(x < - 3\), the point at \(x = - 3\) for this part is an open - circle.

Step2: Analyze the second part of the piece - wise function

For \(y=\frac{5}{3}\sqrt[5]{x}\) when \(x\geq - 3\). When \(x=-3\), \(y=\frac{5}{3}\sqrt[5]{-3}=-\frac{5}{3}\sqrt[5]{3}\approx - 2.1\). So there is a closed - circle at \(x=-3\) for this part.

Answer:

The graph with an open - circle at \(x = - 3\) for the line \(y =-\frac{1}{5}x\) (for \(x < - 3\)) and a closed - circle at \(x=-3\) for the curve \(y=\frac{5}{3}\sqrt[5]{x}\) (for \(x\geq - 3\)) is the correct one. (Since the actual options are not labeled, but based on the analysis of open and closed points at \(x=-3\) for the two sub - functions, the graph can be identified among the given choices)