Problem 20 - Intermediate Algebra

Question

For each function, determine its inverse, \(f^{-1}(x)\) a. \(f(x)=10^{x}\) b. \(f(x)=3^{x}\) c. \(f(x)=\log x\) d. \(f(x)=\log _{2} x\)

Step-by-Step Solution

Answer

a. \(f^{-1}(x) = \log_{10} x\), b. \(f^{-1}(x) = \log_{3} x\), c. \(f^{-1}(x) = 10^{x}\), d. \(f^{-1}(x) = 2^{x}\).

Step 1: Understand the Concept of Inverses

The inverse of a function, denoted as \(f^{-1}(x)\), is a function that "reverses" the effect of the original function \(f(x)\). For a function \(f\), its inverse \(f^{-1}\) satisfies \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 2: Recognizing Inverse Operations

Identify the operations in the function that need to be reversed. For exponential functions, the inverse is a logarithm, and for logarithmic functions, the inverse is an exponential.

Step 3: Find the Inverse of Exponential Functions

For part (a), the function is \(f(x) = 10^{x}\). The inverse of an exponential function \(a^{x}\) is the logarithm to the base \(a\). Thus, \(f^{-1}(x) = \log_{10} x\).For part (b), where \(f(x) = 3^{x}\), the inverse is \(f^{-1}(x) = \log_{3} x\).

Step 4: Find the Inverse of Logarithmic Functions

For part (c), the function is \(f(x) = \log x\). The inverse of the \(\log\)-function (base 10 if not specified) is the exponential function with base 10. So, \(f^{-1}(x) = 10^{x}\).For part (d), \(f(x) = \log_{2} x\). The inverse operation is \(2^{x}\), therefore, \(f^{-1}(x) = 2^{x}\).

Step 5: Write the Final Answer

- a. \(f^{-1}(x) = \log_{10} x\)- b. \(f^{-1}(x) = \log_{3} x\)- c. \(f^{-1}(x) = 10^{x}\)- d. \(f^{-1}(x) = 2^{x}\)

Key Concepts

Exponential Functions

Exponential functions are a key concept in mathematics and occur when a constant base is raised to a variable exponent. This is written as \( a^{x} \), where \( a \) is a positive constant and \( x \) is the exponent. The function describes how the base multiplied by itself for the number of times specified by the exponent. These functions are important because they describe growth and decay processes in various fields such as finance, physics, and biology.

For instance, in the exercise provided, we have functions like \( 10^{x} \) and \( 3^{x} \). Here:

• The base, 10 or 3, refers to the factor that repeatedly multiplies.
• The variable \( x \) denotes the number of times the base is used as a factor.
• These functions grow rapidly as \( x \) increases, characterizing an exponential growth pattern.

Exponential functions have the special property where their inverse is a logarithmic function. This relationship is extremely helpful in solving equations involving exponential expressions.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm of a number is the exponent to which the base must be raised to produce that number. It is represented as \( \, \log_a(x) \), where \( a \) is the base of the logarithm and \( x \) is the number we are taking the log of.

Consider the logarithmic function \( \, \log(x) \) which often implies a base of 10 and \( \, \log_{2}(x) \) with a base of 2 as seen in the examples:

• For \( \, \log(x) \), we find that \( \, 10^{ ext{log}(x)} = x \), reverting back to our original expression in terms of exponential function.
• Likewise, \( \, 2^{ ext{log}_2(x)} = x \).
• Logarithms simplify the multiplication of large numbers, which can then be added as their logs.

Understanding logarithms can be particularly useful in simplifying calculations, solving exponential equations, and analyzing growth in scientific investigations.

Function Operations

Function operations involve various ways of altering and combining functions. Common function operations include addition, subtraction, multiplication, division, and composition of functions. Each of these affects how functions interact and the outcomes produced.

The composition of functions, denoted as \( (f \circ g)(x) = f(g(x)) \), is an operation that applies one function to the results of another. In examining inverse functions, the operation:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

illustrates function composition, showing that applying a function and then its inverse will return the initial value.

This concept is particularly vital in calculus and solving equations, enabling transformations and simplifying complex expressions. Function operations are fundamental in developing a deeper understanding of mathematical relationships and dynamic systems.

Mathematical Inverses

A mathematical inverse of a function reverses the operation of that function, effectively "undoing" its effect. If \( f(x) \) transforms \( x \) in some way, then \( f^{-1}(x) \) will return \( x \) to its original state. This concept is essential for solving equations and understanding functions fully. For any function and its inverse:

• The relationship \( f(f^{-1}(x)) = x \) holds true, making the sequence into a sort of mathematical balancing act.
• Finding the inverse function often involves switching the roles of variables and solving algebraically, such as flipping \( y = f(x) \) into \( x = f(y) \).
• Not all functions have inverses that are also functions (they must pass the Horizontal Line Test).

Understanding inverse functions is crucial because they allow us to solve for variables in the context of real-world applications and provide insights into the structure and symmetry of mathematical models.