Problem 12 - Algebra and Trigonometry

Question

11–14 ? Graph both functions on one set of axes. $$ f(x)=3^{-x} \quad \text { and } \quad g(x)=\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

Graphing both functions results in identical overlapping curves of exponential decay.

Step 1: Understand the Functions

Both functions, \( f(x) = 3^{-x} \) and \( g(x) = \left(\frac{1}{3}\right)^x \), are exponential functions. Notice that the base of \( g(x) \) is \( \frac{1}{3} \), which is the same as \( 3^{-1} \). This means that \( g(x) \) can be rewritten as \( 3^{-x} \). Thus, \( f(x) = g(x) \).

Step 2: Simplify the Functions

From the equality, \( f(x) = 3^{-x} = g(x) \), we can conclude that these two functions are actually identical. Hence, graphing them would result in the same curve.

Step 3: Sketch the Graph

Since \( f(x) = g(x) \), you only need to plot one graph. The function \( 3^{-x} \) is a decreasing exponential function. As \( x \) increases, \( 3^{-x} \) approaches zero but never actually reaches it. As \( x \) decreases, \( 3^{-x} \) shoots towards infinity.

Step 4: Describe the Graph

The graph of \( f(x) = 3^{-x} \) starts high for negative \( x \) values and approaches the x-axis as \( x \) increases to positive values, reflecting an exponential decay. It's a smooth curve with no breaks, passing through the point (0,1) since \( 3^0 = 1 \).

Key Concepts

Graphing Exponential Functions

Exponential functions are intriguing due to their rapid growth or decay. A key feature of these functions is their form: \( y = a^x \), where \( a \) is a positive constant. However, the impact of the base (\( a \)) is profound in determining the behavior of the graph.

In the given exercise, you're asked to graph \( f(x) = 3^{-x} \) and \( g(x) = \left(\frac{1}{3}\right)^x \). It's essential first to understand that any exponential function either grows or decays as \( x \) changes. For \( f(x) \) and \( g(x) \), both express a decay because they involve negative exponents or a fractional base.

The similarity of these functions highlights a critical idea: different forms of representing an exponential can yield the same behavior. Both functions can be reduced to \( 3^{-x} \), simplifying our graphing task.

Exponential Decay

Exponential decay describes a process decreasing rapidly at first, then more slowly over time. It's mathematically modeled by functions where the base of the exponent is between 0 and 1, or expressed with a negative exponent.

This concept is evident in our functions \( f(x) = 3^{-x} \) and \( g(x) = \left(\frac{1}{3}\right)^x \). Both depict an exponential decay as they decrease as \( x \) increases. As \( x \) becomes larger (positive), the value of the function gets closer and closer to zero, without ever reaching it.

Such functions are common in real-world contexts, such as radioactive decay or cooling processes, where the rate of change decreases over time.

Function Equality

Many mathematical problems involve examining if two functions are equal, as seen here with \( f(x) = 3^{-x} \) and \( g(x) = \left(\frac{1}{3}\right)^x \). To determine equality, consider simplifying expressions. As shown, \( \left(\frac{1}{3}\right)^x \) can be rewritten as \( 3^{-x} \), which matches \( f(x) \).

Recognizing these connections is crucial, not just for simplifying math problems, but also for understanding broader mathematical relationships. This process underscores the need to keep all forms flexible, converting and equating expressions to reveal underlying patterns. In this case, it simplifies the graphing task considerably, showing both functions occupying the same space in the graph.

Graphing Techniques

Graphing exponential functions involves a few fundamental techniques that can help you accurately sketch the graph. Start with understanding key points, such as the y-intercept, which is found by setting \( x = 0 \). For \( f(x) = 3^{-x} \), this results in the point (0,1) on the graph.

The behavior as \( x \) approaches positive and negative infinity is crucial. With \( 3^{-x} \), as \( x \) increases, the graph approaches the x-axis, displaying the exponential decay behavior. Conversely, as \( x \) decreases, it climbs steeply towards infinity.

Always ensure you have a smooth curve with no sharp turns or breaks, reflecting how exponential functions change continuously. Use a series of evenly spaced points to facilitate smooth drawing, especially around interesting behaviors like intercepts or asymptotic behavior.