Question

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=-3 e^{-x}$$

Short answer

Answer: The end behaviors of the function f(x) = -3e^(-x) are as follows: As x approaches positive infinity, the function approaches 0, and as x approaches negative infinity, the function approaches negative infinity. There is a horizontal asymptote at y = 0, and there is no vertical asymptote for this function.

Step-by-step solution

Evaluate the limit as x approaches positive infinity

To find the end behavior of the function as x approaches positive infinity, we need to evaluate the limit: $$\lim_{x \to \infty}(-3 e^{-x})$$ Since exponential functions with negative exponents decrease rapidly as x approaches positive infinity, this limit equals to: $$\lim_{x \to \infty}(-3 e^{-x}) = 0$$

Evaluate the limit as x approaches negative infinity

Next, let's evaluate the limit as x approaches negative infinity: $$\lim_{x \to -\infty}(-3 e^{-x})$$ In this case, exponential functions with negative exponent increase rapidly as x approaches negative infinity, so the limit equals: $$\lim_{x \to -\infty}(-3 e^{-x}) = -\infty$$

Identify asymptotes

From the previously calculated limits, we can identify if there are any horizontal or vertical asymptotes in the function. Since the limit as x approaches positive infinity equals 0, we have a horizontal asymptote at y = 0. There is no vertical asymptote for this function as it is continuous for all x.

Sketch the graph

Using the information obtained from the limits, we can now sketch the graph of the function: 1. Draw the horizontal asymptote at y=0. 2. The function decreases from -∞ to 0 as x moves from negative infinity to positive infinity. The resulting graph should start from the bottom left (decreasing rapidly), with the curve approaching the horizontal asymptote at y = 0 as x approaches positive infinity.

Key concepts

End Behavior

End behavior in mathematics refers to the behavior of a function's graph as it approaches either positive or negative infinity. When studying functions, especially transcendental ones, it's crucial to understand how they behave at these extremes, as it can reveal important characteristics about the function.

In this specific exercise, we are dealing with the function \( f(x) = -3 e^{-x} \). To determine its end behavior, we evaluate two limits:

• \( \lim_{x \to \infty}(-3 e^{-x}) \)
• \( \lim_{x \to -\infty}(-3 e^{-x}) \)

As \( x \to \infty \), \( e^{-x} \) decreases rapidly to zero because the function exhibits exponential decay, leading \( f(x) \) to approach 0. Conversely, as \( x \to -\infty \), \( e^{-x} \) increases, making the overall function value approach \(-\infty\). These analyses give us a clear picture of how the function behaves at the extreme ends of the x-axis.

Exponential Functions

Exponential functions form a special class of mathematical functions characterized by a constant base raised to a variable exponent.

For \( f(x) = -3 e^{-x} \), the main body \( e^{-x} \) signifies an exponential decay because \( x \) has a negative exponent. Let's break down what this means:

• **Exponential Growth vs Decay:** In general, an exponential function with a positive exponent grows, while one with a negative exponent (like \( e^{-x} \)) decays.
• **Behavior:** Here, as \( x \) increases, the exponent becomes more negative, and \( e^{-x} \) approaches zero, which is why \( f(x) \) approaches a horizontal asymptote on the x-axis.

Understanding exponential functions is crucial, as they frequently appear in contexts such as natural growth, decay processes, and various scientific models.

Asymptotes

Asymptotes are lines that a graph approaches but never actually meets. They provide insight into the behavior of functions, especially at extreme values of \( x \).

In this function \( f(x) = -3 e^{-x} \), we identified a horizontal asymptote at \( y = 0 \). Here's why:

• **Horizontal Asymptote:** As \( x \to \infty \), the function \( -3 e^{-x} \) gets closer to zero due to the value of \( e^{-x} \) approaching zero. Therefore, the line \( y = 0 \) serves as a boundary that the function nears but never reaches.
• **No Vertical Asymptote:** The function is continuous across all real values of \( x \). This means there are no breaks or undefined points which typically lead to vertical asymptotes.

Understanding asymptotes helps us predict how functions behave far from their starting points. They are integral to sketching accurate graphs and understanding limiting behaviors of functions.