Problem 78 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. Graph the inverse of \(f(x)=3^{x}\)

Step-by-Step Solution

Answer

The inverse of \(f(x) = 3^x\) is \(f^{-1}(x) = \log_3 x\), a logarithmic function passing through (1,0).

Step 1: Identify the Inverse Function

To find the inverse of the function \(f(x) = 3^x\), we need the output to become the input and vice versa. Therefore, we set \(y = 3^x\) and solve for \(x\) in terms of \(y\).

Step 2: Solve for x in Term of y

Express \(x\) in terms of \(y\) by using logarithms. Starting with \(y = 3^x\), take the logarithm base 3 of both sides to get \(x = \log_3 y\). Thus, the inverse function is \(f^{-1}(x) = \log_3 x\).

Step 3: Graph the Inverse Function

Graph the inverse function \(f^{-1}(x) = \log_3 x\). The function is a logarithmic curve, which passes through the point (1,0) on the coordinate plane and approaches negative infinity as \(x\) approaches zero from the positive side. The graph is undefined for non-positive values of \(x\).

Step 4: Verify the Inverse Function

To ensure correctness, verify that the function \(f(x) = 3^x\) and its inverse \(f^{-1}(x) = \log_3 x\) satisfy the property of inverses, where \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. Just like the step-wise approach explained in transforming an exponential equation to its logarithmic form, you start by "solving for the exponent" in an exponential equation. For example, if you have an equation like \(y = 3^x\), converting it into its inverse form utilizes a logarithm with the base of the exponential function. Here, the base is 3, so you derive the inverse as \(x = \log_3 y\). This function can aptly be represented as \(f^{-1}(x) = \log_3 x\).
Logarithmic functions have unique characteristics:

• They pass through the point (1, 0) on a graph, meaning \(\log_b 1 = 0\) for any base \(b\).
• They approach negative infinity as \(x\) approaches zero from the positive side, which means they are undefined for \(x \leq 0\).
• They grow very slowly compared to polynomial or exponential functions.

Understanding the properties and behavior of logarithms is vital to mastering the concept of inverses of exponential functions. By practicing, you can gain deeper insights into how these relate with their exponential counterparts.

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent. For example, consider the function \(f(x) = 3^x\). Here, 3 is the base, and \(x\) is the exponent. The critical role of exponential functions is that they model growth and decay processes, appearing in fields like finance, population dynamics, and physics.
The characteristics of exponential functions include:

• They always cross the y-axis at 1, since any base raised to the power of zero is 1. Therefore \(f(0) = b^0 = 1\).
• They continuously grow larger or smaller depending on whether the base is greater than or less than one, respectively.
• Their rate of increase or decrease accelerates as \(x\) values increase.

When dealing with exponential functions, one of the main aspects of study is converting them to linear forms through logarithms. This conversion is essential because it allows more straightforward computation and analysis, especially when working with inverse functions.

Graphing Inverse Functions

Graphing the inverse of a function involves reflection across the line \(y = x\). When the inverse of an exponential function \(f(x) = 3^x\) is considered, the inverse function \(f^{-1}(x) = \log_3 x\) is reflected in this manner. To successfully graph a logarithmic function, remember these tips:

• Logarithmic functions will cross the x-axis at (1, 0).
• They are undefined for \(x \leq 0\), so ensure you only consider positive x-values.
• The graph is a smooth curve that rapidly approaches negative infinity as it nears zero.

Plotting both the function \(f(x) = 3^x\) and its inverse \(f^{-1}(x) = \log_3 x\) on the same graph allows you to visualize the symmetry which characterizes inverse relationships. You can confirm whether you've graphed correctly by checking that each function reflects over the line \(y = x\), maintaining the vital property that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This in-depth understanding will aid in developing skills to translate complex mathematical relationships into visual representations.