Problem 24 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x-3)}$$

Step-by-Step Solution

Answer

The vertical asymptotes of the graph of the function \(g(x)=\frac{x+3}{x(x-3)}\) are at \(x = 0\) and \(x = 3\).

Step 1: Set the denominator equal to zero

To find the vertical asymptotes we set the denominator of the function, which is \(x(x-3)\), equal to zero. This gives us: \(x(x-3) = 0\)

Step 2: Solve for x

We then solve this equation for \(x\) to find out the points at which the function is undefined. This will result in two values: \(x = 0\) and \(x = 3\).

Step 3: Check the numerator at these points

Then we have to check the numerator at these points. We substitute \(x = 0\) and \(x = 3\) into the numerator \(x+3\). For \(x = 0\), we get \(0+3\) which is nonzero. And for \(x = 3\), we get \(3+3\) which is also nonzero.

Step 4: Establishing the vertical asymptotes

We have seen that the function is not defined for \(x = 0\) and \(x = 3\), and therefore the lines \(x = 0\) and \(x = 3\) are the function's vertical asymptotes.

Key Concepts

Rational Functions

Rational functions are mathematical expressions defined by the ratio of two polynomials. Specifically, they take the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is non-zero. Understanding rational functions provides insight into various aspects of their behavior, including continuity, domain, and asymptotes.

For instance, the function provided in the exercise, \( g(x) = \frac{x+3}{x(x-3)} \) is a rational function. Its domain excludes values of \( x \) that cause the denominator to be zero, as division by zero is undefined in mathematics. Asymptotes, particularly vertical ones, are lines that the graph of the function approaches but never touches. They occur at values of \( x \) where the denominator becomes zero and are crucial for sketching accurate graphs of rational functions.

Undefined Points in Functions

In the world of mathematics, a function may not be able to produce an output for certain inputs—these are known as undefined points. For rational functions, undefined points occur where the denominator equals zero, which corresponds to vertical asymptotes on the graph of the function.

The process of finding these points involves setting the denominator equal to zero and solving for \( x \). In our example, the denominator \( x(x-3) \) when set to zero, yields \( x = 0 \) and \( x = 3 \) as undefined points. This is because if we substitute these values back into \( g(x) \), the expression becomes indeterminate, which means the function does not have a real number output at these points. Identifying undefined points helps in understanding the limitations and special features of the function's domain.

Solving Equations

Solving equations is a fundamental skill in algebra. The process involves finding the values of variables that make the equation true. Equations can be simple linear calculations or more complex ones involving higher-degree polynomials, as seen in the context of rational functions.

In the step-by-step solution, we solve a very basic equation: \( x(x-3) = 0 \). To find solutions, we look for values of \( x \) that satisfy the equation. In this case, using the zero product property allows us to set each factor of the product equal to zero separately, leading to \( x = 0 \) and \( x = 3 \) as solutions. These solutions are critical in the graphing of rational functions because they pinpoint locations of the vertical asymptotes, which are lines that the function approaches but does not cross or touch.