Question

consider the following function.
r(x)=\begin{cases}\frac{5}{8x^{5}}&\text{if }x < 1\\-\frac{3}{8}x&\text{if }x>1end{cases}
step 3 of 3: identify the correct graph of this piece - wise - defined function.

Explanation:

Step1: Analyze \(r(x)=\frac{5}{8x^{5}}\) for \(x < 1\)

When \(x<1\), the function \(y = \frac{5}{8x^{5}}\) is a rational - function. As \(x\to0^{-}\), \(y\to-\infty\); as \(x\to0^{+}\), \(y\to+\infty\). When \(x\) is a large negative number, \(y\to0^{-}\).

Step2: Analyze \(r(x)=-\frac{3}{8}x\) for \(x > 1\)

When \(x > 1\), the function \(y=-\frac{3}{8}x\) is a linear - function with a slope \(m =-\frac{3}{8}\). When \(x = 1\), \(y=-\frac{3}{8}\), and the line extends to the right - hand side of \(x = 1\).

Step3: Consider the non - definition at \(x = 1\)

The function is not defined at \(x = 1\), so there is a break in the graph at \(x = 1\).

Since no options for the graph are provided, we cannot give a specific final answer. But the general characteristics of the graph are as described above. If options were given, we would look for a graph that has a hyperbola - like shape for \(x<1\) (the graph of \(y=\frac{5}{8x^{5}}\)) and a straight - line with a negative slope for \(x > 1\) (the graph of \(y =-\frac{3}{8}x\)) with a break at \(x = 1\).

Answer:

Step1: Analyze \(r(x)=\frac{5}{8x^{5}}\) for \(x < 1\)

When \(x<1\), the function \(y = \frac{5}{8x^{5}}\) is a rational - function. As \(x\to0^{-}\), \(y\to-\infty\); as \(x\to0^{+}\), \(y\to+\infty\). When \(x\) is a large negative number, \(y\to0^{-}\).

Step2: Analyze \(r(x)=-\frac{3}{8}x\) for \(x > 1\)

When \(x > 1\), the function \(y=-\frac{3}{8}x\) is a linear - function with a slope \(m =-\frac{3}{8}\). When \(x = 1\), \(y=-\frac{3}{8}\), and the line extends to the right - hand side of \(x = 1\).

Step3: Consider the non - definition at \(x = 1\)

The function is not defined at \(x = 1\), so there is a break in the graph at \(x = 1\).

Since no options for the graph are provided, we cannot give a specific final answer. But the general characteristics of the graph are as described above. If options were given, we would look for a graph that has a hyperbola - like shape for \(x<1\) (the graph of \(y=\frac{5}{8x^{5}}\)) and a straight - line with a negative slope for \(x > 1\) (the graph of \(y =-\frac{3}{8}x\)) with a break at \(x = 1\).