Problem 9 - Algebra and Trigonometry

Question

\(7-14\) . Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$ y=e^{-x}-1 $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((-1, \infty)\), Asymptote: \(y=-1\).

Step 1: Identify the base function

The given function is a transformation of the base function \( y = e^x \). Recognizing the base function helps us understand how transformations will affect it.

Step 2: Apply transformation to the base function

The function \( y = e^{-x} - 1 \) involves two transformations to \( y = e^x \):1. Reflect \( y = e^x \) over the y-axis to get \( y = e^{-x} \).2. Shift the result downward by 1 unit to get \( y = e^{-x} - 1 \).

Step 3: Sketch the transformed graph

Start with \( y = e^x \) which is an increasing curve. Reflect it over the y-axis to get \( y = e^{-x} \), making it a decreasing curve. Move this entire curve down by 1 unit to obtain the graph of \( y = e^{-x} - 1 \).

Step 4: Determine the domain

The domain of \( y = e^{-x} - 1 \) is the same as that of \( y = e^x \), which is all real numbers. Therefore, the domain is \( (-\infty, \infty) \).

Step 5: Determine the range

Since \( y = e^{-x} \) ranges from (0, \( \infty \)), shifting it down by 1 unit changes the range to \( (-1, \infty) \).

Step 6: Identify the horizontal asymptote

For the function \( y = e^{-x} - 1 \), as \( x \to \infty \), \( e^{-x} \to 0 \). Thus, \( y \to -1 \). The horizontal asymptote is \( y = -1 \).

Key Concepts

Domain and Range

Understanding the domain and range of a function is essential in graphing. The domain refers to all possible input values (or x-values) that a function can accept. For the exponential function \( y = e^{-x} - 1 \), its domain is all real numbers \((-\infty, \infty)\). This is because exponential functions are defined everywhere on the real number line:

• There are no restrictions on the x-values for \( e^{-x} \).
• No operations like division by zero or square roots of negative numbers occur.

The range of a function is all possible output values (or y-values). Initially, for \( y = e^{-x} \), the output values are \((0, \infty)\), as the function approaches zero but never touches it. Shifting the function downward by 1 changes the range:

• The whole graph moves one unit down.
• This adjustment changes the range to \((-1, \infty)\).

This means the smallest value of \( y \) is now just slightly above \(-1\), and it can grow without bound.

Transformations of Exponential Functions

Transformations change the appearance of functions without altering their basic nature. For the function \( y = e^{-x} - 1 \), two transformations are applied to the base exponential function \( y = e^x \):

• First, an exponential reflection occurs over the y-axis, transforming \( y = e^x \) into \( y = e^{-x} \). A reflection across the y-axis affects the x-values by making them negative, which turns the increasing graph of \( e^x \) into the decreasing graph of \( e^{-x} \).
• Second, the function is shifted down by 1 unit, modifying it to \( y = e^{-x} - 1 \). Vertical shifts adjust the y-values directly by adding or subtracting from them. Here, subtracting 1 lowers every point on the graph by one unit, impacting the range and aligning the asymptote at \( y = -1 \).

Such transformations allow us to model real-world situations by adjusting a standard function to fit specific conditions.

Asymptotes

An asymptote is a line that a graph approaches but never actually reaches. In the function \( y = e^{-x} - 1 \), the horizontal asymptote is crucial for understanding the behavior of the graph as \( x \) becomes very large. As \( x \to \infty \), the expression \( e^{-x} \to 0 \), causing the entire function \( y = e^{-x} - 1 \) to approach \( -1 \).

• The line \( y = -1 \) acts like a barrier the function draws near to but never intersects.
• This horizontal asymptote illustrates that despite the diminishing value of \( e^{-x} \), \( y \) will always be slightly greater than \(-1\).

Understanding asymptotes helps in predicting the long-term behavior of functions and in sketching more accurate graphs, as these are the guidelines shaping the curve's path.