Problem 21 - 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. $$ f(x)=\frac{x}{x+4} $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{x}{x+4}\) has a vertical asymptote at \(x=-4\) and has no holes.

Step 1: Identify the vertical asymptotes

The vertical asymptotes of a rational function can be found by setting the denominator equal to zero and solving for \(x\). Therefore, \(x+4=0\). Solving this equation for \(x\) gives \(x=-4\). The function has a vertical asymptote at \(x=-4\).

Step 2: Determine the hole values

Holes in the graph of a rational function occur when there are values that make both the numerator and denominator equal to zero. However, in this function \(f(x)=\frac{x}{x+4}\), there are no values of \(x\) that will make both the numerator \(x\) and the denominator \(x+4\) equal to zero at the same time. Therefore, there are no holes in the graph of this function.

Key Concepts

Understanding Vertical Asymptotes in Rational Functions

In rational functions, vertical asymptotes occur at the values of \(x\) that make the denominator zero, provided these values don't also make the numerator zero, which would instead suggest a hole.
For a function like \(f(x) = \frac{x}{x+4}\), we seek where the denominator, \(x+4\), equals zero. By setting \(x+4=0\) and solving for \(x\), we find the vertical asymptote at \(x=-4\).
The graph of \(f(x)\) will stretch infinitely near this line but will never touch it. Vertical asymptotes represent points of discontinuity, shaping the way the rational function behaves close to these values of \(x\). Understanding their presence and impact is critical to sketching and interpreting the behavior of rational functions.

Exploring Holes in Graphs of Rational Functions

Holes in the graph of a rational function occur when a value of \(x\) makes both the numerator and the denominator zero simultaneously. This results in a point of removable discontinuity.
For example, had the function been \(f(x)=\frac{x(x-2)}{(x+4)(x-2)}\), the \(x-2\) factor appearing in both the numerator and denominator would produce a hole at \(x=2\).
To find a hole in a given rational function, identify common factors between the numerator and denominator after factoring them fully.
However, in our specific example \(f(x)=\frac{x}{x+4}\), there is no \(x\) that makes both parts zero at once, indicating that the graph has no holes.

Why the Denominator Zero is Crucial in Rational Functions

In rational functions, the denominator holds great significance. Setting the denominator to zero helps identify points where the function is undefined.
When a denominator of a rational function, like \(x+4\) in \(f(x)=\frac{x}{x+4}\), is zero, it suggests a possible vertical asymptote unless canceled by a zero in the numerator.

• If these zeros do not cancel each other out, the result is a vertical asymptote.
• If they do cancel out, it creates a hole instead of an asymptote.

Recognizing this distinction is vital for accurate representation and analysis of the function's graph, influencing both the shape and limits of the function across the domain.