Question

Writing to Learn Explain why the equation \(e^{-x}=x\) has at least one solution.

Short answer

The behavior of \(f(x) = e^{-x}\) and \(g(x) = x\) on the interval \([0, +\infty)\) guarantees, according to the intermediate value theorem, that there will be at least one \(x\) where they intersect, providing a solution to the equation \(e^{-x} = x\).

Step-by-step solution

Identify the Functions

Firstly, split the given equation into two separate functions: \(f(x) = e^{-x}\) and \(g(x)=x\).

Analyze the Behavior of the Functions

We'll start analyzing these two functions. Firstly, \(f(x) = e^{-x}\) is a decreasing function in the interval \([0, +\infty)]\), with \(f(0) = 1\) and \(lim_{x \to +\infty} f(x) = 0\). Moreover, \(g(x)=x\) is an increasing function in the interval \([0, +\infty)]\), with \(g(0) = 0\) and \(lim_{x \to +\infty} g(x) = +\infty\).

Use the Intermediate Value Theorem to Prove at Least One Solution Exists

According to the intermediate value theorem, a function that is continuous in the interval \([a, b]\) assumes all the values between \(f(a)\) and \(f(b)\). When \(x=0\), \(f(x)>g(x)\) and when \(x\) goes to \(+\infty\), \(f(x) < g(x)\). Therefore, one can conclude there is a point \(c\) in the interval \((0, +\infty)\) at which \(f(c) = g(c)\). Meaning, there is a solution for the equation \(e^{-x}=x\).

Key concepts

Continuous Functions

Understanding continuous functions is crucial when dealing with equations that involve variables changing smoothly without any breaks or jumps. A continuous function is a mathematical function that, intuitively speaking, has no abrupt changes in its value. In more formal terms, a continuous function at a point 'c' is one where the limit of the function as it approaches 'c' from either direction is equal to the function's value at 'c'.

When applied to real-world scenarios, continuous functions model situations where a sudden change is not possible or observed. For instance, you can think of the temperature during the day as a continuous function—it changes gradually and does not suddenly drop or increase in an instant.

When working with the Intermediate Value Theorem, the requirement that the function be continuous on the interval \[a, b\] ensures that the function takes on all values between \(f(a)\) and \(f(b)\). This is what allows us to confidently claim that a solution to an equation lies within a certain interval, as long as the function remains continuous over that interval.

Exponential Functions

Exponential functions, which are widely used across various scientific disciplines, represent rapid growth or decay. The general form of an exponential function is \(f(x) = a^x\), where 'a' is a positive constant, and the base 'a' is raised to the power of 'x'. Two important properties of exponential functions involve their end behavior and their rate of change.

The function \(e^{-x}\) is a specific type of exponential function. Here, 'e' represents Euler's number, which is approximately 2.71828, and it has unique mathematical properties that make it a natural choice for describing continuous growth and decay processes.

Additionally, all exponential functions are continuous and differentiable everywhere in their domain. This means that we can analyze them using tools like the Intermediate Value Theorem, and they follow a smooth curve without breaks, making them predictable and thus easier to work with when solving equations.

Limits of Functions

In calculus, the concept of limits helps us understand the behavior of functions as they approach a specific value or as 'x' tends toward infinity. Limits allow us to formally define and analyze the notion of continuity. When the limit of a function as 'x' approaches a particular value equals the function's value at that point, we say the function is continuous at that point.

For example, the statement \(\lim_{x \to +\infty} f(x) = 0\) means that as 'x' gets larger and larger, the value of the function \(f(x)\) gets closer and closer to 0. In the context of the exercise \(e^{-x} = x\), analyzing the limits at zero and infinity helps us establish the behavior or trend of the exponential decay relative to the linear growth of 'x', which is essential for determining where the two functions intersect.

Solving Exponential Equations

To solve exponential equations, it's important to understand how to manipulate and compare different exponential expressions. One such technique involves taking the natural logarithm of both sides, which leverages the property that the logarithm of a variable to a base is the exponent that the base must be raised to yield that variable.

However, not all exponential equations can be easily solved with algebra alone. That's where graphical analysis or numerical methods come in. Graphical analysis involves plotting the functions and visually determining their points of intersection. Numerical methods may include iterative approaches like the Newton-Raphson method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

In the given exercise, instead of solving the equation \(e^{-x} = x\) algebraically or numerically, we understand the behavior of both sides of the equation as functions and apply the Intermediate Value Theorem to establish the existence of a solution within a certain interval. This theorem bypasses the nitty-gritty of calculation and gives us a powerful tool to determine the solution's existence.