Problem 117 - Applied Calculus

Question

For each statement, state in words the values of \(x\) for which each exponential expression is defined. $$ x^{-1} $$

Step-by-Step Solution

Answer

The expression \(x^{-1}\) is defined for all real numbers except zero.

Step 1: Understanding the Expression

The given expression is \(x^{-1}\). This expression represents the reciprocal of \(x\), meaning \(x^{-1} = \frac{1}{x}\). Our task is to find the values of \(x\) for which this expression is defined.

Step 2: Identifying Undefined Values

An expression like \(\frac{1}{x}\) is undefined when the denominator is zero. Therefore, we need to find when \(x\) would make the denominator (which is \(x\) itself) equal to zero. Solving \(x = 0\) gives us \(x = 0\). Thus, \(x^{-1}\) is undefined when \(x = 0\).

Step 3: Defining the Value Range

Since the expression is undefined at \(x = 0\), we can conclude that \(x^{-1}\) is defined for all real numbers except zero. Therefore, the values of \(x\) for which \(x^{-1}\) is defined are all real numbers except zero.

Key Concepts

Reciprocal Function

When we talk about a reciprocal function, we are referring to a mathematical function that is the inverse of a given value. Specifically, if you have a number such as \(x\), its reciprocal is expressed as \(x^{-1}\) or \(\frac{1}{x}\). Essentially, this means you are dividing 1 by \(x\).

The concept of reciprocals is important because it helps us understand how division and multiplication are related. For instance, multiplying a number by its reciprocal will always give you 1. This is because \(x \times \frac{1}{x} = 1\).

This is useful when simplifying mathematical expressions or solving equations. Additionally, reciprocal functions showcase their importance in more complex algebraic structures and analysis of functions.

Undefined Values

Any expression becomes undefined when it leads to division by zero. This is a critical aspect of understanding mathematical expressions because division by zero does not yield a meaningful value. In the case of the reciprocal expression \(x^{-1}\) which equates to \(\frac{1}{x}\), it is undefined when \(x = 0\).

Let's break it down:

• When \(x = 0\), the denominator in \(\frac{1}{x}\) becomes zero.
• Division by zero is undefined in mathematics because there is no number that you can multiply by zero to get a non-zero numerator.

Hence, understanding undefined values helps in knowing the limitations and domains of expressions and functions. It ensures that mathematical operations remain valid and logically sound.

Real Numbers

Real numbers are the set of numbers that include all the rational and irrational numbers. They encompass a wide range of values including integers, fractions, and non-repeating decimals. The real number line visualizes these numbers as a continuous line extending infinitely in both positive and negative directions.

With the exponential expression \(x^{-1}\), real numbers play a significant role because this expression is defined for all real numbers except zero. Here's why:

• For any real number \(x\) other than zero, \(\frac{1}{x}\) results in a valid, defined expression.
• The real number set, therefore, serves as the domain for many mathematical functions, except at points where they might lead to undefined operations, such as division by zero.

Understanding real numbers gives us a complete picture of where certain expressions and functions can operate, maintaining mathematical validation across equations.