Problem 22 - A Graphical Approach to Precalculus with Limits

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. $$f(x)=\left(\frac{2}{3}\right)^{x}$$

Step-by-Step Solution

Answer

Domain: (-∞, ∞), Range: (0, ∞); Asymptote: y = 0; f is decreasing on its domain.

Step 1: Determine the Basic Function Type

The function given is an exponential function with the base \( \frac{2}{3} \), which is less than 1. Exponential functions of the form \( a^x \) where \( 0 < a < 1 \) are decreasing.

Step 2: Identify the Domain of the Function

The domain of the function \( f(x) = \left(\frac{2}{3}\right)^{x} \) includes all real numbers, since exponentiation with a positive base is defined for any real number \( x \). Thus, the domain is \(( -\infty, \infty )\).

Step 3: Identify the Range of the Function

Since the function \( \left(\frac{2}{3}\right)^{x} \) is an exponential function that never reaches zero and is always positive, the range of the function is \(( 0, \infty )\).

Step 4: Find the Equation of the Asymptote

For exponential functions of the form \( a^x \), the horizontal asymptote is always the line \( y = 0 \) (the x-axis), because as \( x \to -\infty \), the value of \( f(x) \to 0 \).

Step 5: Determine Whether the Function is Increasing or Decreasing

Since the base \( \frac{2}{3} \) is less than 1, the function \( f(x) = \left(\frac{2}{3}\right)^{x} \) is a decreasing function over its entire domain. As \( x \) increases, \( f(x) \) decreases.

Step 6: Graph the Function

Sketch the graph of \( f(x) = \left(\frac{2}{3}\right)^{x} \) by hand, noting that it's a decreasing curve starting from above the x-axis and approaching it as \( x \to \infty \). Use a graphing calculator to verify this sketch, ensuring it shows \( (0,1) \) as the y-intercept, a decreasing trend, and \( y = 0 \) as the horizontal asymptote.

Key Concepts

Domain and Range

For any function, understanding the domain and range is fundamental. The domain of a function represents all the possible input values, while the range represents all potential output values. In the case of exponential functions like \( f(x) = \left(\frac{2}{3}\right)^{x} \), the domain includes all real numbers. This is because exponentiation with a positive base is valid for every real number \( x \). Thus, we express the domain as \((-\infty, \infty)\).

For the range of exponential functions, the scenario differs. The function \( \left(\frac{2}{3}\right)^{x} \) is always positive and never touches zero, which establishes the range as \((0, \infty)\). No matter the value of \( x \), the output can never be zero or negative in this case.

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

Asymptotes

Asymptotes are lines that a graph approaches but never actually meets. For exponential functions of the form \( a^x \), a very common type of asymptote is the horizontal asymptote. In our specific function \( f(x) = \left(\frac{2}{3}\right)^{x} \), the horizontal asymptote is the x-axis, which is expressed as the line \( y = 0 \).

This happens because as \( x \to -\infty \), the function \( f(x) \) approaches zero but never truly reaches it. Essentially, an asymptote clarifies the behavior of a function at the extreme ends of its domain, offering a clear indicator of how the graph behaves as it continues infinitely.

• Horizontal asymptote: \( y = 0 \)

Increasing and Decreasing Functions

Identifying whether a function is increasing or decreasing helps in understanding its behavior over an interval. An exponential function like \( f(x) = \left(\frac{2}{3}\right)^{x} \) has unique characteristics because its base is less than 1. Such functions typically decrease as \( x \) increases.

In this context, as \( x \) becomes larger, \( f(x) \) diminishes, illustrating that \( f(x) \) is a decreasing function. The broader mathematical rule is that if the base of an exponential function is between 0 and 1, the function is decreasing over its entire domain.

• The function is decreasing because \( \frac{2}{3} < 1 \).
• As \( x \to \infty \), \( f(x) \to 0 \).

Graphing Techniques

Graphing an exponential function can begin with identifying key characteristics such as the y-intercept and the horizontal asymptote. For \( f(x) = \left(\frac{2}{3}\right)^{x} \), the y-intercept occurs where \( x = 0 \), so \( f(0) = 1 \), giving us the point (0,1). This is a crucial anchor point for sketching the graph.

Then, you might want to choose a few more values of \( x \) and compute \( f(x) \) to understand the curve's behavior, especially confirming its decreasing nature. Since this is a decreasing function, you'll start plotting from the y-intercept, and as \( x \) increases, \( f(x) \) will approach but not reach the line \( y = 0 \), the horizontal asymptote.

• Start point: Y-intercept at \( (0, 1) \)
• Plot additional points for better accuracy.
• Ensure the curve approaches the asymptote without touching it.

Using a graphing calculator can help verify your hand-drawn graph, ensuring accurate depiction of these features.