Problem 37 - Algebra for College Students

Question

\(f(x)=\frac{x}{|x|}\)

Step-by-Step Solution

Answer

\( f(x) = 1 \) for \( x > 0 \) and \( f(x) = -1 \) for \( x < 0 \); undefined for \( x = 0 \).

Step 1: Understand the function definition

The function is defined as follows: \( f(x) = \frac{x}{|x|} \). Here, \(|x|\) represents the absolute value of \(x\).

Step 2: Consider different values of x

Since absolute value varies based on the sign of \(x\), we consider two cases: (1) when \(x > 0\) and (2) when \(x < 0\).

Step 3: Evaluate for x > 0

If \(x > 0\), then \(|x| = x\). Substituting in the function, \( f(x) = \frac{x}{x} = 1\).

Step 4: Evaluate for x < 0

If \(x < 0\), then \(|x| = -x\). Substituting in the function, \( f(x) = \frac{x}{-x} = -1\).

Step 5: Evaluate for x = 0

The function is not defined for \(x = 0\) because division by zero is undefined. So, \( f(x) \) does not exist for \(x = 0\).

Step 6: Write the final piecewise function

Based on the evaluation, the piecewise function can be expressed as: \[ f(x) = \begin{cases} 1 & \text{if } x > 0 \ -1 & \text{if } x < 0 \end{cases} \]

Key Concepts

Absolute Value

The absolute value of a number is like its distance from zero on the number line. It shows how far the number is from zero, regardless of direction. This means:

• If a number is positive, its absolute value is the number itself.
• If a number is negative, its absolute value is the number without the negative sign.

In mathematical terms, for a number \(x\), its absolute value is denoted as \(|x|\). Thus:

• \(|x| = x\) if \(x > 0\)
• \(|x| = -x\) if \(x < 0\)
• \(|x| = 0\) if \(x = 0\)

Understanding absolute value is crucial when analyzing piecewise functions, which can change based on whether values are positive or negative.

Division by Zero

Division by zero is a mathematical no-go because it doesn't have a meaningful result. When you divide a number by another, you're essentially distributing the first number into parts as indicated by the second number. But when that second number is zero, it's like dividing something non-existent into parts - which can't happen. Hence:

• Any number divided by zero is undefined.
• No mathematical operation can create a valid number in this situation.

In the given function \(f(x) = \frac{x}{|x|}\), when \(x = 0\), this leads to trying to divide by zero, making \(f(x)\) undefined at \(x = 0\). This is why \(x = 0\) isn't included in the piecewise solution.

Function Evaluation

Function evaluation involves plugging different values into a function to find corresponding outputs. For our function \( f(x) = \frac{x}{|x|} \), evaluation varies depending on the value and sign of \(x\). Let's break it down:

• For \(x > 0\), \(|x| = x\), so the function becomes \( f(x) = \frac{x}{x} = 1 \).
• For \(x < 0\), \(|x| = -x\), so the function becomes \( f(x) = \frac{x}{-x} = -1 \).

The evaluation requires examining cases where the absolute value changes the sign or leaves it untouched. Understanding each situation helps determine how to proceed with the function as a whole.

Mathematical Cases

Mathematical cases refer to evaluating different segments of a function based on varying conditions, like particular intervals of \(x\). For the provided function, it involves these cases:

• \(x > 0\): Where the function simplifies to return \(1\).
• \(x < 0\): Where the function simplifies to return \(-1\).

It’s important to be methodical when setting these cases to avoid mistakes. This is especially true in piecewise functions, where different rules apply depending on the situation. As observed, for \(x = 0\), the problem is undefined due to division by zero. Case evaluations ensure that each condition is properly handled, providing a clear understanding of what the function returns at various points.