Problem 84 - Precalculus : Building Concepts and Connections

Question

Explain why the function \(f(x)=2^{x}\) has no vertical asymptotes (review Section 4.6).

Step-by-Step Solution

Answer

The function \(f(x) = 2^{x}\) has no vertical asymptotes because it doesn't tend towards either negative or positive infinity as its inputs approach any real number.

Step 1: Understand the function

Let's start by understanding the given function, \(f(x) = 2^{x}\). This function is an exponential function with base 2. By definition, the exponential function \(f(x) = a^{x}\), where a is a positive number not equal to 1, is defined for all real numbers x. That means the domain of \(f(x)\) is \(-\infty, +\infty\).

Step 2: Observing the behavior of the function

Next, let's observe the behavior of \(f(x)\). As a function with positive base greater than 1, \(f(x)=2^{x}\) increases as x increases. But more importantly, it never drops below 0. As x tends towards -\infty, \(f(x)\) approaches 0, and as x tends towards +\infty, \(f(x)\) goes to infinity. But at no point does it behave in such a way as to cause \(f(x)\) to go to either negative or positive infinity as it approaches a specific real number.

Step 3: Conclusion

Based on these observations, we can say that the function \(f(x) = 2^{x}\) has no vertical asymptotes. There are no values of x for which the function tends towards either negative or positive infinity, which means there is no vertical line to which the graph will get infinitely close without touching or crossing.

Key Concepts

Exponential Function

An exponential function, like the one in our example, can be represented as \( f(x) = a^x \), where the base \( a \) is a positive real number different than 1. When the base is greater than one, the exponential function will always exhibit growth as the value of \( x \) increases. The interesting quality of such functions is that they grow rapidly - much faster than linear or quadratic functions.

An important property of exponential functions is that they are never equal to zero. No matter how negative the value of \( x \) becomes, an exponential function with a base greater than 1 approaches zero but never actually reaches it. This concept is especially important when discussing the behavior of the function and understanding why it does not possess vertical asymptotes.

Function Behavior

The behavior of a function describes how the output values change in response to the input values. For the exponential function \( f(x) = 2^x \), as \( x \) becomes more positive, the function value gets larger because the base, 2, is raised to higher powers. Conversely, as \( x \) becomes more negative, \( f(x) \) gets closer and closer to zero, never becoming negative or reaching zero itself.

This behavior is crucial for understanding why there are no vertical asymptotes in this case. A vertical asymptote occurs when the function approaches infinity or negative infinity as \( x \) approaches a particular value. However, since our function approaches a finite value (zero) and goes to infinity only as \( x \) goes to positive infinity (which is not a specific value), we conclude that no vertical asymptotes are present.

The smooth and predictable behavior of the exponential function over its entire domain implies a graph that extends infinitely in the horizontal direction without any sudden jumps or vertical spikes that would suggest the presence of a vertical asymptote.

Domain of a Function

The domain of a function is the set of all possible input values (\( x \) values) for which the function is defined. For our exponential function \( f(x) = 2^x \), the domain is all real numbers, which is mathematically represented as \( -fty, fty \). This means you can substitute any real number into \( f(x) \), and you will get a meaningful result out of the function.

The significance of the domain in relation to vertical asymptotes lies in understanding that vertical asymptotes can only occur at values that are not included in the domain. Since our function has an all-encompassing domain, it logically follows that there are no values at which the function is not defined and consequently, no vertical asymptotes can exist.