Problem 23 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ g(x)=\frac{x+3}{x(x+4)} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function \(g(x)=\frac{x+3}{x(x+4)}\) are at \(x = 0\) and \(x = -4\). The function has no holes.

Step 1: Identify the function's denominator

The first step is to identify the denominator of the function. The denominator of the function \(g(x)\) given here is \(x(x+4)\).

Step 2: Find the vertical asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for \(x\). In this case, that gives \(x(x+4) = 0\) which satisfy \(x = 0\) or \(x = -4\).

Step 3: Determine the values causing a hole

Now we need to check if these are actual vertical asymptotes, or if any of them is a hole. This involves solving the nominator equally zero. For a \(x\) value to result in a hole, both nominator and denominator need to be zero. The equation \(x+3=0\) yields \(x=-3\), which is not equal to the solutions of step 2, proving that \(x = 0\) and \(x = -4\) are indeed vertical asymptotes without holes.

Key Concepts

Holes in Rational Functions

In rational functions like the one given, holes occur where both the numerator and the denominator are zero at the same x-value. A hole is essentially a point on the graph where the function does not exist, though it would appear smooth and continuous otherwise. To find holes, you need to:

• Set the numerator of the function to zero and solve for \(x\).
• Compare these \(x\) values with those from the denominator set to zero.

If an \(x\)-value is shared between these calculations, it indicates a hole. In our case, the numerator \((x+3)\) equals zero at \(x=-3\), but the denominator \(x(x+4)\) does not share this zero, thereby confirming no holes occur in this scenario.

Denominator Polynomial

The denominator polynomial of a rational function is crucial for understanding its domain and the behavior of the graph. The irregularities in the graph, like vertical asymptotes and holes, emanate from this part of the function. It forms the foundation for identifying any restrictions on \(x\):

• To identify these restrictions, set the denominator equal to zero.
• Solve the resulting equation to uncover where the function is undefined.

In our exercise, the denominator \(x(x+4)\) expands to \(x^2 + 4x\), which simplifies finding where the function equals zero, namely where \(x=0\) or \(x=-4\). These are the potential vertical asymptotes or possibly holes.

Graphical Analysis

Graphical analysis of rational functions involves studying the curve to understand its behavior and key features. Important elements are:

• Vertical asymptotes – the lines where the function approaches infinity.
• Holes – detached points in the graph.
• Domain restrictions – derived from where the denominator equals zero.

For our function \(g(x)\), the vertical asymptotes \(x=0\) and \(x=-4\) suggest the graph has undefined points at these locations where it typically approaches infinity. While performing a graphical analysis, plotting these points and observing the behavior near these critical \(x\)-values can deepen understanding.

Finding Asymptotes

Finding asymptotes in rational functions is key to understanding the graph's structure. Vertical asymptotes occur where the denominator equals zero and the function approaches infinity. To find these, common steps include:

• Set the denominator equal to zero: \(x(x + 4) = 0\).
• Solve to find potential asymptote \(x\)-values: \(x=0\) and \(x=-4\).

Verify that these solutions are not also zeros of the numerator to confirm them as asymptotes. Unlike horizontal asymptotes that describe end-behavior, vertical asymptotes like \(x=-4\) and \(x=0\) show local behavior changes, indicating where the graph shoots up to infinity.