Problem 73 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=\log _{2}(x-1)\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = 2^x + 1\).

Step 1: Understand the function

The given function is \(f(x) = \log_2(x - 1)\). This is a logarithmic function with base 2.

Step 2: Write the function in terms of y

To find the inverse, first express the function as \(y = \log_2(x - 1)\). This equation helps us switch x and y in the next step.

Step 3: Switch x and y

Replace x with y and y with x to get the equation \(x = \log_2(y - 1)\). This is the first step to finding the inverse.

Step 4: Solve for y

To isolate y, we change the logarithmic equation to its exponential form: \(y - 1 = 2^x\).

Step 5: Finalize the inverse function

Solve for y by adding 1 to both sides: \(y = 2^x + 1\). The inverse function is \(f^{-1}(x) = 2^x + 1\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions. They help us to determine the power to which a given base number must be raised to obtain another number. Logarithmic functions are written in the form \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. For example, in \( f(x) = \log_2(x - 1) \), the base is 2, and it shows us what power 2 must be raised to, in order to result in \( x - 1 \).
Logarithms have several important properties that make them useful in various mathematical operations. Here are a few key properties:

• The product rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• The quotient rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• The power rule: \( \log_b(m^n) = n \cdot \log_b(m) \)

These properties are crucial when expanding or simplifying logarithmic expressions.
In the process of finding inverse functions, like \( f^{-1}(x) \) for a logarithmic function, we need to switch the x and y values and then solve the resulting equations. This is achieved by converting the logarithmic form of the equation into its equivalent exponential form.

Exponential Functions

Exponential functions are mathematical expressions in the form \( f(x) = b^x \), where \( b \) is a positive real number and the base of the exponent. They are powerful tools for modeling real-world applications such as population growth, radioactive decay, and interest calculations. Understanding exponential functions aids in solving inverse problems involving logarithms, because each logarithmic expression can be re-written in an exponential form.
For example, if we have a logarithmic expression \( y = \log_2(x-1) \), we can convert it into an exponential form by solving for x as an exponent: \( x - 1 = 2^y \). This switch is crucial for finding the inverse of the given logarithmic function. When re-arranging it to solve for y in terms of x, we get \( y = 2^x + 1 \), completing the inverse transformation.

• Exponential growth: It occurs when a quantity increases rapidly over time and follows the general pattern \( y = a \cdot b^x \), where \( a \) is the initial amount.
• Exponential decay: This is when a quantity decreases rapidly, expressed as \( y = a \cdot b^{-x} \) or \( y = a \cdot (\frac{1}{b})^x \).

Mastering these conversions between exponential and logarithmic forms is essential for working with inverse functions.

Function Transformations

Function transformations modify the basic graph of a function in various ways such as shifting, stretching, compressing, or reflecting it. Understanding these transformations helps to identify and work with different forms of the same function effectively.
For instance, in the original function \( f(x) = \log_2(x - 1) \), the 'x - 1' part indicates a horizontal shift. Specifically, it shifts the graph of the function 1 unit to the right along the x-axis.

• Horizontal shifts: Represented by \( f(x) = \log_b(x - h) \), moving the graph right or left by \( h \).
• Vertical shifts: Given by \( f(x) = \log_b(x) + k \), moving up or down by \( k \).
• Reflections: Across the x-axis or y-axis, they change the graph's orientation, e.g., \( -f(x) \).
• Stretches/Compressions: Affect the graph's size without altering its shape, influenced by multiplying by a constant factor.

Each transformation has a specific impact on the function's graph and understanding them aids in graphically solving and interpreting inverse functions, like transitioning from a logarithmic expression to its inverse exponential form.