Question

consider the table representing a rational function.which statement describes the graph of the function?the function has holes when ( x = 0 ), ( x = 3 ), and ( x = 5 ).the function has vertical asymptotes when ( x = 0 ), ( x = 3 ), and ( x = 5 ).the function has holes when ( x = 0 ) and ( x = 3 ), and a vertical asymptote when ( x = 5 ).the function has a hole when ( x = 3 ), and vertical asymptotes when ( x = 0 ) and ( x = 5 ).

Explanation:

Step1: Analyze the behavior near \(x = 0\)

As \(x\) approaches \(0\) from the left (\(x=-0.1,-0.01,-0.001\)), \(f(x)\) gets very large (\(1.96,19.96,199.96\)). As \(x\) approaches \(0\) from the right (\(x = 0.001,0.01,0.1\)), \(f(x)\) gets very large in the negative direction (\(-200.04,-20.04,-2.04\)). This is the behavior of a vertical - asymptote.

Step2: Analyze the behavior near \(x = 3\)

As \(x\) approaches \(3\) from the left (\(x = 2.9\)), \(f(x)=-0.16\) and as \(x\) approaches \(3\) from the right (\(x = 3.1\)), \(f(x)=-0.17\). The function values are approaching a finite value near \(x = 3\). When a function is undefined at a point \(x=a\) but the limit as \(x\to a\) exists, there is a hole at \(x = a\).

Step3: Analyze the behavior near \(x = 5\)

As \(x\) approaches \(5\) from the left (\(x = 4.9,4.99\)), \(f(x)\) gets very large in the negative direction (\(-2.04,-20.04\)). As \(x\) approaches \(5\) from the right (\(x=5.001,5.01,5.1\)), \(f(x)\) gets very large (\(199.96,19.96,1.96\)). This is the behavior of a vertical - asymptote.

Answer:

The function has a hole when \(x = 3\), and vertical asymptotes when \(x = 0\) and \(x = 5\).