Problem 14 - Intermediate Algebra

Question

For what value(s) of \(x\) is each function undefined? a. \(f(x)=\frac{x-7}{x}\) b. \(\quad f(x)=\frac{x+1}{x-3}\) c. \(f(x)=\frac{x^{2}-2}{x(x+8)}\) d. \(f(x)=\frac{8 x}{(x-1)(x+1)}\)

Step-by-Step Solution

Answer

a: undefined at \(x = 0\); b: \(x = 3\); c: \(x = 0, -8\); d: \(x = 1, -1\).

Step 1: Identify Undefined Conditions

For any function in the form of a fraction, the function is undefined when its denominator is equal to zero. We'll need to identify the values of \(x\) that make the denominator zero for each function.

Step 2: Check Function a

Function \(f(x) = \frac{x-7}{x}\) is undefined when the denominator \(x = 0\). Therefore, the function is undefined at \(x = 0\).

Step 3: Check Function b

Function \(f(x) = \frac{x+1}{x-3}\) is undefined when the denominator \(x-3 = 0\). Solving for \(x\), we get \(x = 3\). Hence, the function is undefined at \(x = 3\).

Step 4: Check Function c

Function \(f(x) = \frac{x^2-2}{x(x+8)}\) is undefined when the denominator \(x(x+8) = 0\). Setting each factor of the denominator to zero, we find \(x = 0\) and \(x = -8\). Thus, the function is undefined at \(x = 0\) and \(x = -8\).

Step 5: Check Function d

Function \(f(x) = \frac{8x}{(x-1)(x+1)}\) is undefined when the denominator \((x-1)(x+1) = 0\). Solving \(x-1 = 0\) and \(x+1 = 0\) gives \(x = 1\) and \(x = -1\). Therefore, the function is undefined at \(x = 1\) and \(x = -1\).

Key Concepts

Rational Functions

Rational functions are a type of mathematical function that involves ratios of polynomials. This means they take the form of a fraction where both the numerator and the denominator are polynomial expressions. A simple example can be something like \(f(x) = \frac{2x + 3}{x - 5}\). These functions are interesting because they can exhibit features like asymptotes and discontinuities, which are points where the function is not defined.

Understanding how rational functions behave is crucial in algebra and calculus, as they form the building blocks for more complex concepts in mathematics.

• The numerator determines the roots or zeroes of the function, which are the values of \(x\) that make the numerator \(0\).
• The denominator dictates the undefined points, as these are the values that make the denominator \(0\).

Recognizing these singularities in rational functions helps in graphing them and predicting their behavior over different intervals.

Denominators

In any fraction, the denominator is the term below the division line. For rational functions, the denominator is of utmost importance as it dictates where the function may not be continuous. When dealing with rational functions, always pay close attention to the denominator since setting it equal to zero will help identify points of discontinuity.

Here's why denominators matter:

• A denominator of zero cannot exist in real number arithmetic because division by zero is undefined. This makes such values critical to identify in rational functions.
• By finding the values excluded by the denominator, we understand where a function might "break" or become undefined.

To find these critical points, simplify the denominator using algebraic operations to isolate the variables, subsequently calculating their values. Keeping an eye on the denominator ensures you accurately plot and assess the behavior of rational functions.

Zero Denominator

The concept of a zero denominator is central to understanding where rational functions become undefined. Specifically, a rational function is undefined at any point where its denominator equals zero. This is because division by zero is not permissible in standard mathematics, creating what we refer to as an undefined value or point of discontinuity.

Here’s a step-by-step approach to dealing with zero denominators:

• Identify the denominator of the function, such as in \(f(x) = \frac{1}{x+2}\), where the denominator is \(x+2\).
• Set the denominator equal to zero, i.e., \(x+2 = 0\).
• Solve the resulting equation to find the value(s) of \(x\) that make the denominator zero. In this example, it’s \(x = -2\).
• Conclude that the function is undefined at \(x = -2\).

This method is vital for safely handling rational functions and avoiding erroneous calculations. Understanding where these zeros occur helps in accurate graphing and analysis.

Undefined Values

When a rational function gives an undefined value, it means there's a point where the function cannot produce a valid output. This typically occurs at places where the function's denominator is zero. The undefined values are essential in characterizing the function's domain—the set of all possible \(x\)-values for which the function produces a valid output.

It's essential to identify and appropriately handle these undefined values:

• They help you determine the domain of the function by excluding these problematic \(x\)-values.
• Undefined values often correspond to vertical asymptotes on the graph of the function, which are vertical lines that the graph approaches but never touches or intersects.
• Providing restrictions in real-world problems, indicating limitations on input values.

Embracing the concept of undefined values is a significant step in understanding the nature of functions and is crucial for accurately solving both theoretical and practical problems. Knowing these points ensures clarity in mathematical communication and problem-solving.