Problem 60 - Elementary and Intermediate Algebra

Question

How do the graphs of \(f(x)=3^{x}\) and \(g(x)=\left(\frac{1}{3}\right)^{x}\) differ? How eare they similar?

Step-by-Step Solution

Answer

The graphs of \(f(x)\) and \(g(x)\) differ in growth behavior: \(f(x)\) grows, while \(g(x)\) decays. They are similar in y-intercept, domain, and range.

Step 1: Identify Base of Functions

First, identify the base of each function. The function \(f(x) = 3^x\) has a base of 3, while the function \(g(x) = \left(\frac{1}{3}\right)^x\) has a base of \(\frac{1}{3}\). The bases are reciprocals of each other.

Step 2: Determine Growth Behavior

In an exponential function, if the base is greater than 1, the function exhibits exponential growth. Therefore, \(f(x) = 3^x\) demonstrates exponential growth as \(x\) increases. Conversely, if the base is between 0 and 1, the function exhibits exponential decay. Hence, \(g(x) = \left(\frac{1}{3}\right)^x\) shows exponential decay.

Step 3: Compare Y-Intercepts

Both functions intersect the y-axis at the same point. When \(x = 0\), \(f(x) = 3^0 = 1\) and \(g(x) = \left(\frac{1}{3}\right)^0 = 1\). Therefore, both functions have a y-intercept at \( (0, 1) \).

Step 4: Analyze Symmetry

The graphs of \(f(x)\) and \(g(x)\) are reflections of each other across the y-axis. This is because the change in base from 3 to \(\frac{1}{3}\) reflects the point (\(x, y\)) to (\(-x, y\)).

Step 5: Sketch and Compare Domains and Ranges

Both functions have the same domain, \((-\infty, +\infty)\), meaning they can take any real value for \(x\). Their ranges are also the same, \((0, +\infty)\), meaning they only produce positive outputs.

Key Concepts

Exponential Growth and Decay

Exponential functions are powerful mathematical expressions that model how quantities grow or decay over time. They appear in numerous real-world contexts, such as population growth, radioactive decay, and interest calculations.

In general, an exponential function can be represented as \( f(x) = a^x \), where \( a \) is the base. Whether the function shows exponential growth or decay depends on the value of the base:

• If the base \( a \) is greater than 1, the function represents exponential growth. As \( x \) increases, the function \( f(x) \) grows rapidly.
• If the base \( a \) is between 0 and 1, the function represents exponential decay. As \( x \) increases, \( f(x) \) decreases towards zero, but never reaches it.

In our exercise, \( f(x) = 3^x \) portrays exponential growth because the base is 3. Meanwhile, \(g(x) = \left(\frac{1}{3}\right)^x\) shows decay because the base \( \frac{1}{3} \) is between 0 and 1.

Reciprocal Bases

The concept of reciprocal bases arises when analyzing how similar bases can influence the shape and behavior of exponential functions. A reciprocal is essentially the flipped version of a number. Therefore, for a base \( a \), its reciprocal is \( \frac{1}{a} \). This means if you have \( a^x \), its reciprocal base is \( (\frac{1}{a})^x \).

In our exercise, \( f(x) = 3^x \) and \( g(x) = \left(\frac{1}{3}\right)^x \) are prime examples of functions with reciprocal bases. This reciprocal relationship results in reflection properties on the graph.

When bases are reciprocals:

• The growth function \( a^x \) and the decay function \( (\frac{1}{a})^x \) mirror each other across the y-axis in a graph.
• They share the same y-intercept. Both \( f(x) = 3^x \) and \( g(x) = \left(\frac{1}{3}\right)^x \) intersect the y-axis at \( (0, 1) \).

This symmetry emphasizes the interconnected nature of exponential growth and decay.

Graph Transformations

Graph transformations allow us to visually understand and interpret the impact of different parameters in functions on their graphical representation. With exponential functions, slight changes in the equations can cause notable variations in the corresponding graphs.

In the specific case of \( f(x) = 3^x \) and \( g(x) = \left(\frac{1}{3}\right)^x \), graph transformations take place due to:

• Reciprocal Bases: These functions are reflections across the y-axis, meaning at point \( (x, y) \) on one graph, you have a point \( (-x, y) \) on the other graph.
• Identical Y-intercepts: Both functions intersect the y-axis at \( (0, 1) \), showcasing that no matter the base (as long as it maintains the reciprocal relationship), they have a shared intersection point.
• Similar Domains and Ranges: Both functions can take any real number for \( x \) (domain of \( (-\infty, +\infty) \)), and always yield positive outputs (range of \( (0, +\infty) \)). This reflects their essential nature, but with different orientations depending on the base's magnitude.

By understanding these key transformations, learners can predict and graph exponential functions correctly, recognizing the nuances that different bases introduce.