Problem 23 - A Graphical Approach to College Algebra

Question

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \((0, \infty)\); Asymptote: \(y=0\); \(f\) is increasing.

Step 1: Understand the Function

The function given is the exponential function \( f(x) = e^x \). This function is continuous and smooth, and it shows exponential growth as \( x \) increases.

Step 2: Identify the Domain

For the function \( f(x) = e^x \), the domain is the set of all real numbers. This means \( x \) can take any value from negative infinity to positive infinity, written as \( (-\infty, \infty) \).

Step 3: Identify the Range

The range of \( f(x) = e^x \) consists of all positive real numbers because the exponential function never takes a negative value or zero. Hence, the range is \( (0, \infty) \).

Step 4: Determine the Asymptote

The function \( f(x) = e^x \) has a horizontal asymptote along the line \( y = 0 \), which the graph approaches as \( x \) approaches negative infinity.

Step 5: Determine If the Function is Increasing or Decreasing

An exponential function \( f(x) = e^x \) is always increasing, as the derivative \( f'(x) = e^x > 0 \) for all \( x \). This indicates that the function is increasing on its entire domain.

Step 6: Sketch the Graph

Draw the graph of \( f(x) = e^x \) which starts near zero for low values of \( x \), passes through the point \( (0, 1) \), and increases rapidly as \( x \) becomes positive. Using a calculator can aid in confirming critical points and the rapid increase of the graph.

Key Concepts

Domain and Range

When analyzing an exponential function like \( f(x) = e^x \), it's important to understand both its domain and range. The domain refers to all possible input values for \( x \) that the function can accept. For \( e^x \), \( x \) can be any real number, meaning the domain is \((-\infty, \infty)\). This wide domain is typical for exponential functions.

The range, however, is slightly more restricted. Because \( e^x \) produces only positive values (never zero or negative), the range is \((0, \infty)\). This trait is key to many applications of exponential functions, where outputs typically must remain positive.

Here are a few important points about the domain and range of \( f(x) = e^x \):

• Domain: All real numbers \((-\infty, \infty)\)
• Range: Positive real numbers \((0, \infty)\)

Asymptotes

An asymptote is a line that a graph approaches but never actually reaches. For the exponential function \( f(x) = e^x \), there is a horizontal asymptote. This is a key feature to consider when sketching the graph or understanding the behavior of the function as \( x \) takes very large negative values.

For \( e^x \), the horizontal asymptote is the line \( y = 0 \). As \( x \) approaches negative infinity, \( e^x \) gets closer and closer to zero without ever actually reaching it. This behavior is crucial for comprehending how the function behaves particularly for large negative values of \( x \).

Keep in mind about \( f(x) = e^x \):

• Asymptote: Horizontal at \( y = 0 \)

Increasing and Decreasing Functions

When determining if an exponential function is increasing or decreasing, the derivative provides valuable information. For \( f(x) = e^x \), the derivative \( f'(x) = e^x \) is always positive. This indicates that \( e^x \) increases continuously and smoothly as \( x \) increases.

This consistent increase is a defining characteristic of exponential growth. As a result, \( f(x) = e^x \) is considered to be an increasing function across its entire domain. Understanding whether a function is increasing or decreasing helps in predicting how the function's values change over both short and long intervals.

Important aspects of \( f(x) = e^x \):

• Always Increasing: Because \( f'(x) = e^x > 0 \) for all \( x \).