Problem 26 - 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. $$f(x)=e^{x}-1$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((-1, \infty)\), Asymptote: \(y = -1\), and \(f(x)\) is increasing.

Step 1: Analyze the Function

The function given is \( f(x) = e^x - 1 \). This is an exponential function shifted downward by 1 unit. The basic exponential function \( e^x \) has been modified.

Step 2: Determine the Domain

The domain of exponential functions like \( e^x \) is all real numbers, \( (-\infty, \infty) \). The transformation \( -1 \) does not affect the domain, so the domain of \( f(x) = e^x - 1 \) is also \( (-\infty, \infty) \).

Step 3: Determine the Range

The range of \( e^x \) is \( (0, \infty) \). Since \( f(x) = e^x - 1 \), the whole range is shifted down by 1. Therefore, the range of \( f(x) \) is \( (-1, \infty) \).

Step 4: Identify the Asymptote

The horizontal asymptote of \( e^x \) is \( y = 0 \). After shifting the graph vertically down by 1 unit, the new horizontal asymptote is \( y = -1 \).

Step 5: Determine Increasing or Decreasing

An exponential function like \( e^x \) is always increasing. The transformation \( -1 \) only affects vertical position, not increasing behavior. Therefore, \( f(x) = e^x - 1 \) is increasing on its entire domain \( (-\infty, \infty) \).

Step 6: Sketch the Graph

To sketch the graph, begin with the standard \( e^x \) curve. Shift the entire curve down by 1 unit. Ensure the graph approaches but never reaches \( y = -1 \).

Step 7: Verify with Calculator

Use a graphing calculator to confirm the sketch of \( f(x) = e^x - 1 \). Check that the graph approaches the horizontal asymptote \( y = -1 \) and increases steadily.

Key Concepts

Domain and Range

When dealing with exponential functions like \( f(x) = e^x - 1 \), understanding the domain and range is crucial for graphing and analyzing the function's behavior. The *domain* refers to all possible values for \( x \). For the exponential function \( e^x \), and consequently for \( e^x - 1 \), the domain is all real numbers, denoted as \( (-\infty, \infty) \). This means that neither exponential growth nor the transformation affects what \( x \) values you can input.
The *range*, however, is altered by transformations. For the standard \( e^x \), which only outputs positive numbers, the range is \( (0, \infty) \). But since we've subtracted 1 in \( f(x) = e^x - 1 \), each output is shifted downward by 1. Thus, the range changes to \( (-1, \infty) \).

• The domain remains \( (-\infty, \infty) \).
• The range, after shifting, is \( (-1, \infty) \).

This shift is crucial to understanding where the graph lies on the y-axis.

Graphing Functions

Graphing \( f(x) = e^x - 1 \) involves both understanding the base function and how transformations affect it. Start by considering the graph of \( e^x \), which passes through the point (0,1) and increases exponentially. Now, the transformation \( -1 \) means shifting the entire graph downward by one unit.
When graphing by hand:

• Start with a sketch of \( e^x \).
• Shift each point down 1 unit.
• The result should approach a horizontal line at \( y = -1 \) and continue to rise as \( x \) increases.

This visual helps understand both the behavior and placement on a graph. Graphing calculators can confirm your sketches, ensuring accuracy. With practice, visualizing these transformations becomes intuitive.

Asymptotes

Asymptotes are lines that a graph approaches yet never quite touches. For exponential functions like \( e^x - 1 \), identifying them helps in understanding graph limits and direction.
The original function \( e^x \) has a horizontal asymptote at \( y = 0 \). The subtraction of 1 affects this. The asymptote moves down by 1 unit, resulting in a new asymptote at \( y = -1 \).

• The asymptote provides a boundary that the exponential graph approaches but doesn't cross.
• This helps inform graph behavior and range limits.

Recognizing where an asymptote lies helps you predict the long-term behavior of the function as \( x \) values increasingly deviate from zero, particularly for exponential functions.

Function Behavior

Understanding the behavior of \( f(x) = e^x - 1 \) involves recognizing how the function changes with increasing \( x \). Exponential functions like \( e^x \) are inherently *increasing*, meaning as \( x \) gets larger, \( f(x) \) also gets larger.
Even with the transformation \( -1 \), this fundamental behavior of increasing values remains unchanged. The output might start at \(-1\) when \( x = 0 \) but continues to rise as \( x \) increases. This characteristic indicates:

• For any interval within the domain \((-\infty, \infty)\), the function is increasing.
• This increasing trend is a signature aspect of exponential functions.

Knowing that \( f(x) = e^x - 1 \) consistently increases helps in recognizing how the graph's slope behaves and why it doesn't decrease or flatten.