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Chapter 8: Problem 37

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

Short Answer

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

Step by step solution

01

Understand the function definition

The function is defined as follows: \( f(x) = \frac{x}{|x|} \). Here, \(|x|\) represents the absolute value of \(x\).
02

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 \(x < 0\).
03

Evaluate for x > 0

If \(x > 0\), then \(|x| = x\). Substituting in the function, \( f(x) = \frac{x}{x} = 1\).
04

Evaluate for x < 0

If \(x < 0\), then \(|x| = -x\). Substituting in the function, \( f(x) = \frac{x}{-x} = -1\).
05

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 \(x = 0\).
06

Write the final piecewise function

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

If \(y\) is inversely proportional to the square of \(x\), and \(y=\frac{1}{8}\) when \(x=4\), find \(y\) when \(x=8 . \quad y=\frac{1}{32}\)

For each of the following, predict the general shape and location of the graph, and then use your calculator to graph the function to check your prediction. (a) \(f(x)=\sqrt{x^{2}}\) (b) \(f(x)=\sqrt{x^{3}}\) (c) \(f(x)=\left|x^{2}\right|\) (d) \(f(x)=\left|x^{3}\right|\)

A retailer has a number of items that she wants to sell and make a profit of \(40 \%\) of the cost of each item. The function \(s(c)=c+0.4 c=1.4 c\), where \(c\) represents the cost of an item, can be used to determine the selling price. Find the selling price of items that cost \(\$ 1.50\), \(\$ 3.25, \$ 14.80, \$ 21\), and \(\$ 24.20\).

The distance that a freely falling body falls varies directly as the square of the time it falls. If a body falls 144 feet in 3 seconds, how far will it fall in 5 seconds? 400 feet

\(f(x)=\left\\{\begin{aligned} x & \text { for } x \geq 0 \\ 3 x & \text { for } x<0 \end{aligned}\right.\)
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