Problem 79 - Algebra 2

Question

Find the vertical asymptotes of each function. $$ y=\frac{x-3}{x(x-1)} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function \(y=\frac{x-3}{x(x-1)}\) are \(x = 0\) and \(x = 1\).

Step 1: Identify the denominator

The philosophy to find vertical asymptotes is based on the idea that these occur where the function is not defined. Hence, the denominator of the function must be identified, which is \(x(x-1)\) in this case.

Step 2: Set the denominator equal to zero

An asymptote occurs where the function is undefined, which happens when the denominator equals zero. We want to find all x-values for which this is the case. Therefore, set the denominator equal to zero and solve for x: \(x(x-1) = 0\).

Step 3: Solve for x

The equation \(x(x-1) = 0\) can be factored into \(x = 0\) and \(x = 1\). These are the x-values for which the denominator of the function equals zero, meaning the function is undefined at these x-values.

Key Concepts

Rational Functions

Rational functions are expressions that can be represented as the quotient of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). This type of function is crucial in mathematics because it represents relationships that can be described as a division of two quantities.

• The numerator \( P(x) \) governs the behavior of the function's values.
• The denominator \( Q(x) \) determines where the function might become undefined.

Rational functions are very useful because they often model situations in real life, such as rate problems and inverse relationships.
Understanding these functions is essential to grasp advanced mathematical concepts and graphing behaviors.

Function Undefined

A function becomes undefined when you cannot assign a finite value to it for certain inputs. This usually occurs when the denominator of a rational function equals zero, making division impossible.

• Carefully check denominators to identify when they reach zero.
• If the denominator equals zero at a certain point, the function cannot produce a valid output at that point.

In our exercise, we determined that the function \( y = \frac{x-3}{x(x-1)} \) is undefined at points where \( x(x-1) = 0 \).
This forms the basis for identifying any vertical asymptotes associated with the function.

Denominator Zero

The point where the denominator in a rational function is zero is critical, as it indicates the location of vertical asymptotes. Solving the equation formed by setting the denominator to zero gives these specific points. In our case:

• The denominator is \( x(x-1) \), and setting it equal to zero forms the equation \( x(x-1) = 0 \).
• The critical values of \( x \, = 0 \) and \( x \, = 1 \) emerge from this equation.

At these points, \( x = 0 \) and \( x = 1 \), the function does not have defined values due to the denominator being zero.
These are essential locations on the graph where vertical asymptotes exist, highlighted by undefined behavior.

Factoring

Factoring is a crucial skill in solving algebraic equations, especially rational functions. It involves rewriting a polynomial as a product of its simplest components. This process helps uncover vital values that define the function's behavior. For our example:

• The denominator \( x(x-1) \) is already nicely factored into \( x \) and \( (x-1) \).
• By setting each factor equal to zero, we identify key points: \( x = 0 \) and \( x = 1 \).

These identified values from factoring help locate where the function is undefined.
Factoring is beneficial not just in solving problems, but also in simplifying expressions to make analysis more straightforward.