Problem 78 - Elementary and 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 \( f^{-1}(x) = \log_3{x} \). Graph \( f^{-1}(x) = \log_3{x} \) as a curve starting from (1, 0) upwards to the right.

Step 1: Understand the Task

To solve this exercise, we need to first find the inverse of the function \( f(x) = 3^x \), and then graph it. Recall that finding an inverse involves switching the roles of \( x \) and \( y \), and then solving for \( y \).

Step 2: Swap Variables for Inverse

Rewrite the function as \( y = 3^x \). To find the inverse, swap \( x \) and \( y \), giving us \( x = 3^y \).

Step 3: Solve for y

To isolate \( y \), take the logarithm with base 3 of both sides. This gives us \( y = \log_3{x} \). Therefore, the inverse function is \( f^{-1}(x) = \log_3{x} \).

Step 4: Graph the Inverse Function

Recall that \( f(x) = 3^x \) is an exponential function, which is always increasing and passes through the point (0, 1). Its inverse, \( f^{-1}(x) = \log_3{x} \), is a logarithmic function. It will pass through the point (1, 0) and is increasing as well. Plot a curve that approaches the y-axis but never touches it, starting from the x-axis and moving up to the right.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable appears in the exponent. These functions have the general form \(f(x) = a^x\), where \(a\) is a positive constant. The base \(a\) is a crucial part of the equation and determines the curve's behavior. Exponential functions have several key properties:

• They grow very quickly or decay rapidly, depending on whether the base is greater or less than one.
• The graph passes through the point \((0,1)\) since any number raised to the power of zero is one.
• The function never touches the x-axis, indicating that it never results in a zero value.
• These functions are always increasing if \(a > 1\) or decreasing if \(0 < a < 1\).

In the context of our exercise, \(f(x) = 3^x\) is an exponential function with a base of three. This means it rises sharply as \(x\) increases, illustrating the property of exponential growth.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. If \(f(x) = a^x\) is an exponential function, then \(f^{-1}(x) = \log_a(x)\) is the corresponding logarithmic function. The logarithm base \(a\) essentially "undoes" the exponential effect. Key characteristics of logarithmic functions include:

• They pass through the point \((1,0)\), because the logarithm of one at any base is zero.
• The graph increases slowly and approaches the vertical asymptote at the y-axis but never touches or crosses it. This corresponds to the fact that a logarithm of zero is undefined.
• Logarithms convert multiplication into addition; hence, they are incredibly useful for solving equations involving powers of numbers.

For our specific example, the inverse of \(f(x) = 3^x\) is \(f^{-1}(x) = \log_3{x}\). By plotting this graph, we observe it through its unique shape, which highlights the distinct nature of logarithmic growth compared to exponential.

Graphing Functions

Graphing mathematics functions helps in visualizing their behavior. With exponential and logarithmic functions, the relationship is particularly intriguing because they are inverses.When graphing \(f(x) = 3^x\):

• The curve rises sharply to the right, representing exponential growth.
• It passes through the point \((0,1)\) and continues upwards without ever touching the x-axis.

For the inverse function \(f^{-1}(x) = \log_3{x}\):

• The graph starts near the x-axis, passing through \((1,0)\) and moving slowly upwards to the right.
• The y-axis acts as a boundary that the curve approaches but never crosses.

Together, the graphs visually illustrate how one is the mirror image of the other concerning the line \(y=x\). This line represents where the values of \(x\) and \(y\) are equal, highlighting the defining feature of inverse functions.