Problem 12 - Elementary and 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. \( x = 0 \), b. \( x = 3 \), c. \( x = 0, -8 \), d. \( x = 1, -1 \)

Step 1: Understanding Undefined Functions

A function is undefined when the denominator of a rational expression equals zero because division by zero is not possible. Our task is to determine the values of \( x \) that make the denominator zero for each part of the problem.

Step 2: Solving Part (a)

For \( f(x) = \frac{x-7}{x} \), the denominator is \( x \). Set \( x = 0 \) to find when the function is undefined. Therefore, the function is undefined when \( x=0 \).

Step 3: Solving Part (b)

In \( f(x) = \frac{x+1}{x-3} \), the denominator is \( x-3 \). Set \( x-3 = 0 \) to find the undefined point, giving \( x = 3 \). Thus, the function is undefined when \( x=3 \).

Step 4: Solving Part (c)

For \( f(x) = \frac{x^2 - 2}{x(x+8)} \), the denominator is \( x(x+8) \). Set each factor to zero: \( x = 0 \) and \( x+8 = 0 \) (which means \( x = -8 \)). Hence, the function is undefined at \( x = 0 \) and \( x = -8 \).

Step 5: Solving Part (d)

In \( f(x) = \frac{8x}{(x-1)(x+1)} \), the denominator is \( (x-1)(x+1) \). Set each factor to zero: \( x-1 = 0 \) (which gives \( x = 1 \)) and \( x+1 = 0 \) (which gives \( x = -1 \)). Thus, the function is undefined at \( x = 1 \) and \( x = -1 \).

Key Concepts

Rational Expressions

Rational expressions are expressions that can be written as the ratio of two polynomials. These expressions resemble fractions, where both the numerator and the denominator are polynomial expressions. Describing a rational expression can be as simple as having a single variable in the numerator and denominator, or it can be more complex with multiple terms and powers. Key features of rational expressions:

• The numerator and the denominator are both polynomials.
• They can often be simplified just like numerical fractions.
• Rational expressions can sometimes be undefined if conditions are not met, such as when division by zero occurs in the denominator.

Rational expressions are valuable in mathematics because they offer a structured way to approach relationships between variables akin to how fractions compare numbers. Analyzing rational expressions involves ensuring the denominator never equals zero, which leads us to consider undefined functions.

Division by Zero

In mathematics, dividing by zero is a situation that must be avoided because it leads to undefined outcomes. Division by zero does not produce a meaningful result since there is no number that any number can be divided by to result in zero. Why is division by zero undefined?

• Attempting to divide by zero does not yield a real number or a rational result.
• It creates equations that do not satisfy customary arithmetic rules.
• This undefined nature prevents mathematical operations from proceeding logically.
• The undefined results can play havoc in calculations, leading to incorrect conclusions.

Avoiding division by zero is crucial when dealing with rational expressions. A function becomes undefined at points where its expression results in a division by zero, requiring special attention to ensure valid outcomes.

Denominator Equals Zero

For rational expressions, when the denominator equals zero, the expression becomes undefined. This is one of the fundamental rules that need to be adhered to when working with these types of expressions. Implications of a zero denominator:

• If any part of the denominator becomes zero, the entire expression cannot be evaluated.
• Mathematically, you solve for these points by equating the denominator to zero and solving for the variable(s).
• Identifying these points is essential to understand where a function is not defined.

By finding where the denominator equals zero, typically through setting each factor of the denominator to equal zero individually, one can determine the "problem spots" where the function fails to produce a real number. These are key concepts for evaluating and graphing rational functions and understanding their domain. Always pay special attention to these scenarios to avoid pitfalls in solving mathematics problems.