Problem 26 - Applied Calculus

Question

Graph each function. $$ f(x)=\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = \left(\frac{1}{3}\right)^x \) is a decreasing exponential curve with a horizontal asymptote at \( y = 0 \).

Step 1: Recognize the Function Type

The given function \( f(x) = \left(\frac{1}{3}\right)^x \) is an exponential function. Exponential functions have the form \( a^x \), where \( a > 0 \) and \( a eq 1 \). In this case, the base \( a \) is \( \frac{1}{3} \).

Step 2: Determine the Domain and Range

The domain of an exponential function \( f(x) = a^x \) is all real numbers, \( (-\infty, \infty) \). The range for this function, since \( \left(\frac{1}{3}\right)^x \) will always be positive, is \( (0, \infty) \).

Step 3: Identify Horizontal Asymptote

For \( f(x) = \left(\frac{1}{3}\right)^x \), as \( x \to \infty \), \( f(x) \to 0 \), which means there is a horizontal asymptote at \( y = 0 \).

Step 4: Calculate Key Points

Calculate a few points to help plot the graph. For example:\ - When \( x = 0 \), \( f(x) = \left(\frac{1}{3}\right)^0 = 1 \)\ - When \( x = 1 \), \( f(x) = \left(\frac{1}{3}\right)^1 = \frac{1}{3} \)\ - When \( x = -1 \), \( f(x) = \left(\frac{1}{3}\right)^{-1} = 3 \)

Step 5: Plot the Graph

Use the points from Step 4 and the horizontal asymptote from Step 3 to sketch the graph. The function \( f(x) = \left(\frac{1}{3}\right)^x \) is decreasing as \( x \) increases because the base \( \frac{1}{3} \) is less than 1. The graph will pass through the points calculated and approach the horizontal asymptote \( y = 0 \) as \( x \to \infty \).

Key Concepts

Function Graphing

Graphing functions makes them easier to understand visually. For exponential functions like \( f(x) = \left( \frac{1}{3} \right)^x \), graphing helps reveal their behavior over various values of \( x \).
To graph, you need:

• Identification of key points
• Plotting of the points
• Understanding of direction of the curve

This particular exponential function is decreasing because its base is less than 1. As \( x \) increases, \( f(x) \) becomes smaller.
Plotting exponential functions typically results in a curve that smoothly bends and approaches an asymptote. In this case, it means getting closer to the x-axis but never actually touching it.

Horizontal Asymptote

A horizontal asymptote is a line that a graph approaches as \( x \) tends towards infinity. For \( f(x) = \left(\frac{1}{3}\right)^x \), the horizontal asymptote is \( y = 0 \).
As \( x \to \infty \), the exponential expression \( \left(\frac{1}{3}\right)^x \) approaches zero. This is what generates the asymptote at \( y = 0 \). It’s important to understand that the function will get very close to this line but won't actually reach it.
Recognizing horizontal asymptotes helps predict the function's behavior and graph it with better accuracy, providing insights on how the function behaves at extreme values of \( x \).

Domain and Range

Understanding the domain and range of a function is crucial for graphing.
For exponential functions like \( f(x) = \left(\frac{1}{3}\right)^x \):

• Domain: All real numbers (\( -\infty, \infty \)). This means you can input any real number for \( x \), and the function produces a valid output. Exponential functions are defined everywhere on the number line.
• Range: Positive real numbers (\( 0, \infty \)). The function outputs only positive numbers since \( \left( \frac{1}{3} \right)^x \) is always greater than zero for any real \( x \).

This important feature helps you understand the limit and extent of the graph visually.

Plotting Points

Plotting points is essential in creating the graph of a function. It gives discrete reference points for sketching the curve.
For \( f(x) = \left(\frac{1}{3}\right)^x \), let's calculate a few fundamental points:

• At \( x = 0 \), \( f(x) = 1 \) (since any number to the power of zero is 1) - this is where the graph intersects the y-axis.


• At \( x = 1 \), \( f(x) = \frac{1}{3} \) - shows how quickly the function decreases.


• At \( x = -1 \), \( f(x) = 3 \) because the reciprocal of \( \frac{1}{3} \) raised to positive 1 is 3 - indicating that as \( x \) becomes negative, the graph rises.

These points start to outline the shape of the graph which then smoothly continues in a curve reflecting the function's behavior.