Problem 72 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=3^{x+1}\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \log_3(x) - 1 \).

Step 1: Identify the Function

We start with the function given: \[ f(x) = 3^{x+1} \]Our goal is to find the inverse of this function, denoted as \( f^{-1}(x) \). An inverse function essentially reverses the effect of the original function.

Step 2: Replace f(x) with y

To indicate that the function holds an equality, we replace \( f(x) \) with \( y \):\[ y = 3^{x+1} \] This is a crucial step to solve for \( x \) in terms of \( y \), which will help us find \( f^{-1}(x) \).

Step 3: Solve for x in terms of y

Now, we'll solve the equation \( y = 3^{x+1} \) for \( x \). First, take the logarithm base 3 of both sides to eliminate the exponent:\[ \log_3(y) = x + 1 \] Next, solve for \( x \) by isolating it on one side:\[ x = \log_3(y) - 1 \]

Step 4: Express the Inverse Function

With \( x = \log_3(y) - 1 \), we now replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \) to express the inverse function:\[ f^{-1}(x) = \log_3(x) - 1 \] This is the inverse function.

Step 5: Verify the Inverse Function

To ensure that \( f^{-1}(x) \) is correct, check that applying \( f \) and then \( f^{-1} \) returns the original value:1. Calculate \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = 3^{\log_3(x) - 1 + 1} = x \]2. Calculate \( f^{-1}(f(x)) \): \[ f^{-1}(3^{x+1}) = \log_3(3^{x+1}) - 1 = x + 1 - 1 = x \]Both checks show the function and its inverse correctly reverse each other's effect.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable is in the exponent. In the function \( f(x) = 3^{x+1} \), the base is 3, and the exponent is \(x+1\). This means that as the variable \( x \) changes, the value of the function increases or decreases exponentially. There are a few key characteristics of exponential functions:

• Growth Rate: Exponential functions grow by constant percentages, rather than constant differences. As \( x \) increases, \( f(x) \) grows faster compared to linear or quadratic functions.
• Base Effects: The base (in this case, 3) affects the rate at which the function grows. Different bases result in different rates of growth or decay.
• Horizontal Asymptote: As \( x \) approaches negative infinity, \( f(x) \) approaches zero, but never actually reaches it.

Understanding these characteristics is crucial for identifying and solving problems involving exponential functions, especially when dealing with their inverses.

Logarithms

Logarithms are the inverse of exponential functions. They take the form \( \log_b(a) \), which answers the question, "To what power must the base \( b \) be raised, to produce \( a \)?" In the context of our original function \( f(x) = 3^{x+1} \), taking the logarithm base 3 is a strategic move to solve for \( x \) when the function itself has \( x \) in its exponent.Logarithms are vital in unraveling exponential expressions:

• Inverse Relationship: The logarithmic and exponential functions are inverse operations. Applying a logarithmic operation can cancel out an exponential function and vice versa.
• Properties of Logarithms: There are several rules, such as \( \log_b(mn) = \log_b(m) + \log_b(n) \) and \( b^{\log_b(a)} = a \), which simplify calculations involving logarithms.
• Solving Equations: By taking the logarithm of both sides, as shown in the solution, we turn complex exponential equations into manageable linear equations.

These features make logarithms particularly powerful tools in mathematics for functions, especially when determining the inverse.

Function Verification

Function verification is a vital step in ensuring the accuracy of an inverse function. When we find the inverse of a function, such as \( f(x) = 3^{x+1} \), it is essential to check that this inverse accurately reverses the original function.The process of function verification involves two main steps:

• Substitution and Simplification: Calculate \( f(f^{-1}(x)) \). Substitute the inverse into the original function and simplify. For our inverse, \( f^{-1}(x) = \log_3(x) - 1 \), this is confirmed by showing that \( 3^{\log_3(x) - 1 + 1} = x \).
• Reverse Substitution: Calculate \( f^{-1}(f(x)) \). Substitute the function into its inverse and simplify. For \( f(x) = 3^{x+1} \), this step involves showing \( \log_3(3^{x+1}) - 1 = x \).

Both of these calculations should result in the term \( x \), demonstrating that each function effectively undoes the other. Through these checks, we validate that the found inverse function is indeed correct and reliable.