Problem 9 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{7}{2 x} $$

Step-by-Step Solution

Answer

The rational expression is undefined for \( x = 0 \).

Step 1: Understand When a Rational Expression is Undefined

A rational expression is undefined when its denominator is equal to zero. In this exercise, we need to determine for which value of \( x \) the denominator of the given rational expression \( \frac{7}{2x} \) becomes zero.

Step 2: Set the Denominator Equal to Zero

Take the denominator of the expression \( 2x \) and set it equal to zero:\[2x = 0\]

Step 3: Solve for x

Solve the equation from Step 2 for \( x \). Divide both sides of the equation by 2 to isolate \( x \):\[x = \frac{0}{2} = 0\]

Step 4: Conclusion

The value \( x = 0 \) causes the expression \( \frac{7}{2x} \) to be undefined because it results in division by zero.

Key Concepts

Understanding Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Think of them as similar to numerical fractions, but instead of numbers, they use variable expressions. Just like with numerical fractions, rational expressions can sometimes encounter problems when their denominators include variables that make them undefined. When working with rational expressions, it's important to pay attention to the values which make the denominator zero, as those are the points where the expression becomes undefined.

For example, consider the expression \(\frac{7}{2x}\). Here, the numerator is "7" and the denominator is "2x". Despite the simplicity of this expression, it’s crucial to identify which values of \(x\) make the denominator zero, turning the expression undefined.

The Role of the Denominator Zero

The denominator plays a crucial role in determining where a rational expression is undefined. This is because division by zero is undefined in mathematics, creating a critical limitation for rational expressions. To find where these limitations occur, evaluate where the denominator of a given rational expression equals zero. This means finding the values of the variable that result in a zero denominator.

Taking the expression \( \frac{7}{2x} \) as an example, focus on the denominator "2x". To find where this denominator becomes zero, solve the equation \(2x = 0\). Solving this tells us which value of \(x\) causes the denominator to equal zero, which in this case is \(x = 0\).

Avoiding Division by Zero

Division by zero is undefined because there's no number that can result from dividing any number by zero. It leads to an infinite or indeterminate number, which is not allowed in the realm of real numbers. This is why, in expressions like \(\frac{7}{2x}\), identifying the points where division by zero occurs is crucial.

For the expression \(\frac{7}{2x}\), if \(x\) equals zero, then the expression attempts to divide 7 by 0, which is undefined. Hence, as a critical rule of thumb in mathematics: always check the denominator of a rational expression for values that would equal zero when solving or simplifying rational expressions. Avoiding these values ensures that the expression remains valid and defined.