Problem 11 - College Algebra with Modeling and Visualization

Question

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Take the reciprocal of a nonzero number \(x\).

Step-by-Step Solution

Answer

Original: If \(x\neq 0\), then \(y = \frac{1}{x}\). Inverse: If \(y = \frac{1}{x}\), then \(x\neq 0\).

Step 1: Understanding the original statement

The original statement is: 'Take the reciprocal of a nonzero number \(x\)'. This means for any nonzero number \(x\), you find its reciprocal, which is \(\frac{1}{x}\).

Step 2: Define the original statement symbolically

Symbolically, the given statement can be expressed as: If \(xeq 0\), then \(y = \frac{1}{x}\).

Step 3: Understanding the inverse statement

To find the inverse, we need to swap the hypothesis and conclusion. The verbal inverse of the statement is: 'A number whose reciprocal is \(y\) is a nonzero number.'

Step 4: Expressing the inverse symbolically

Symbolically, the inverse statement of the given statement can be expressed as: If \(y = \frac{1}{x}\), then \(xeq 0\).

Key Concepts

Reciprocal

Let's start by understanding what a "reciprocal" is. A reciprocal of a number is simply its "multiplicative inverse." This means when you multiply a number by its reciprocal, you get 1. For any number other than zero, its reciprocal can be found. For example:

• The reciprocal of 2 is \(\frac{1}{2}\).
• The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• The reciprocal of -5 is \(\frac{-1}{5}\).

Remember, zero does not have a reciprocal because dividing by zero is undefined!
Notice that finding reciprocals is very helpful in mathematical computations, especially when solving equations involving fractions.

Symbolic Statements

In mathematics, expressing statements symbolically allows for more precise communication. When we have a verbal statement like "Take the reciprocal of a nonzero number \(x\)," we can express it using symbols for clarity. Here's how it works:

• The original statement involves a condition and a result. The condition is having a nonzero number \(x\), and the result is obtaining its reciprocal \(y = \frac{1}{x}\).
• Symbolically, this can be written as: If \(x eq 0\), then \(y = \frac{1}{x}\).

Using symbolic notation makes it easier to manipulate mathematical statements, allowing one to derive more complex concepts, such as functions and equations.

Inverse Statements

Inverse statements are a fascinating aspect of logical reasoning. To create an inverse statement, you swap the hypothesis and conclusion of the original statement.

• If the original were: "If it rains, I will carry an umbrella," the inverse would be: "If I carry an umbrella, then it rains."
• In our reciprocal example, the symbolic inversion swaps the roles of \(x\) and \(y\), stating: "If \(y = \frac{1}{x}\), then \(x eq 0\)."

Remember, an inverse does not always maintain the truth value of the original statement!
Analyzing inverse statements helps deepen understanding of logical structure and their implications.