Problem 91 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why the equation \(2 e^{x}=-1\) has no solution.

Step-by-Step Solution

Answer

The equation \(2 e^{x}=-1\) has no solution, as the exponential function \(e^{x}\) can never be negative.

Step 1: Understanding the Exponential Function

Let's consider the function \(f(x) = e^{x}\). For all real numbers \(x\), the range of an exponential function, such as \(e^{x}\), is \((0, \infty)\). In simpler terms, an exponential function always produces an output greater than zero, regardless of what real number it is given as an input.

Step 2: Applying the Property of Exponential Function to the Given Equation

The equation provided is \(2 e^{x}=-1\). When we isolate \(e^{x}\) on one side of the equation by dividing both sides by 2, we get \(e^{x}=-0.5\). But as discussed in step one, \(e^{x}\) cannot be negative because the range is \((0, \infty)\). Therefore, the equation \(2 e^{x}=-1\) has no solution as the exponential term \(e^{x}\) can never be negative, so any equation that requires it to be is unsolvable within the real number system.

Key Concepts

Range of Exponential Functions

The range of exponential functions, such as \( e^{x} \), is crucial to understanding why certain equations have no solutions. An exponential function like \( f(x) = e^{x} \) grows rapidly and its output is always positive. This means the range, or set of possible output values, is \((0, \infty)\).
This range indicates that regardless of the input \( x \), \( e^{x} \) will never output a number less than or equal to zero.

• Exponential functions asymptotically approach zero but never reach or drop below it.
• This continuous rise is why you will only see outputs greater than zero.

For our particular equation example, \( 2 e^{x} = -1 \), we see that expecting \( e^{x} \) to equal a negative number is outside the realm of what its range allows. Thus, equations requesting negative solutions from an exponential function have no valid solutions in real numbers.

Properties of Real Numbers

Understanding the properties of real numbers is vital when solving mathematical equations. Real numbers include all the numbers you can find on the number line, from negative infinity to positive infinity, except for imaginary numbers.
These numbers are represented using decimals or fractions.
Real numbers have several important properties, like:

• They are closed under addition, subtraction, multiplication, and division (except by zero).
• They are consistent and follow the order of operations and algebraic rules.

Each real number is unique and follows the established rules of arithmetic. However, some equations may ask for outputs that cannot exist within the real numbers, like a negative result from an exponential function. These instances highlight why understanding both the nature of functions and their interaction with the real numbers is essential.

Solving Equations

Solving an equation involves finding a value that makes the equation true, adhering strictly to mathematical principles. It is crucial to identify the type of function and use the corresponding rules.
In equations with exponential functions, recognize that certain values are not feasible due to the function's inherent properties.

• First, isolate the exponential component, if possible, by dividing or factoring.
• Then, assess the isolated expression based on known ranges or limits.

In the equation \(2 e^{x} = -1\), isolating \( e^{x} \) gives us \( e^{x} = -0.5 \). Recognizing that \( e^{x} \) cannot be negative, we quickly determine there is no real number solution.

Dealing with equations effectively requires an understanding of both the function's behavior and the real number properties that may restrict solutions.