Question

Which of the following is NOT a possible value of \( \frac{1}{4 - x} \)

• A. -4
• B. 4/17
• C. 0
• D. 4
• E. 17/4

Hint:

A fraction of the form \( \frac{k}{f(x)} \), where \(k\) is a non-zero constant, can never be equal to zero. The only way a fraction can be zero is if its numerator is zero.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question deals with the properties of rational functions (fractions with variables). A key property of any fraction \( \frac{a}{b} \) is that its value is zero if and only if the numerator \(a\) is zero and the denominator \(b\) is not zero. Another key property is that the denominator \(b\) cannot be zero.

Step 2: Key Formula or Approach:
Let the given expression be equal to a value \(y\): \[ y = \frac{1}{4 - x} \] We need to determine if there is any value that \(y\) cannot take. The expression is a fraction with a constant numerator (1). For a fraction to be equal to zero, its numerator must be zero.

Step 3: Detailed Explanation:
The numerator of the fraction \( \frac{1}{4 - x} \) is 1. Since the numerator is a constant value of 1, it can never be equal to 0. Therefore, the entire fraction can never be equal to 0, regardless of the value of \(x\) in the denominator.
Let's check if other values are possible by solving for \(x\): If \( y = \frac{1}{4-x} \), then \( y(4-x) = 1 \), which means \( 4-x = \frac{1}{y} \), and \( x = 4 - \frac{1}{y} \).
This equation can be solved for \(x\) for any non-zero value of \(y\). For example: \[\begin{array}{rl} \bullet & \text{If \(y = -4\), \(x = 4 - \frac{1}{-4} = 4.25\).} \\ \bullet & \text{If \(y = 4/17\), \(x = 4 - \frac{17}{4} = -0.25\).} \\ \bullet & \text{If \(y = 4\), \(x = 4 - \frac{1}{4} = 3.75\).} \\ \bullet & \text{If \(y = 17/4\), \(x = 4 - \frac{4}{17} \approx 3.76\).} \\ \end{array}\] All these values are possible. However, if we try to set \(y=0\): \[ 0 = \frac{1}{4 - x} \] To solve this, we would multiply both sides by \(4-x\), resulting in \(0 \times (4-x) = 1\), which simplifies to \(0=1\). This is a contradiction, which means there is no value of \(x\) that can make the expression equal to 0.
Step 4: Final Answer
The value that is NOT possible for the expression is 0.