Problem 83 - Algebra and Trigonometry

Question

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$ f(x)=\log _{2}\left(\log _{10} x\right) $$

Step-by-Step Solution

Answer

(a) Domain: \(x > 1\). (b) Inverse: \(f^{-1}(x) = 10^{2^x}\).

Step 1: Find Domain of Inner Function

The inner function in this composition is \( \log_{10} x \). The argument of a logarithm must be positive, so \( x > 0 \). This gives us the domain for \( \log_{10} x \), restricting \( x \) to be greater than 0.

Step 2: Find Domain of Outer Function

The outer function is \( \log_2(y) \) where \( y = \log_{10} x \). Again, the argument of the logarithm must be positive, so we need \( \log_{10} x > 0 \). This condition implies that \( x > 1 \).

Step 3: Determine Overall Domain

Combine the constraints from Steps 1 and 2. Since the condition from Step 2 \( x > 1 \) is more restrictive than \( x > 0 \), the domain of the entire function is \( x > 1 \).

Step 4: Set Up Equation for Inverse Function

To find the inverse function, we first replace \( f(x) \) with \( y \): \( y = \log_2(\log_{10} x) \). Next, we aim to solve for \( x \) in terms of \( y \).

Step 5: Solve for \(x\)

Re-write the equation \( y = \log_2(\log_{10} x) \) as \( 2^y = \log_{10} x \) by exponentiating each side relative to base 2. This gives \( \log_{10} x = 2^y \). Then, solve for \( x \) by rewriting the equation as \( x = 10^{2^y} \).

Step 6: Write the Inverse Function

Replace \( y \) with \( f^{-1}(x) \): the inverse function is \( f^{-1}(x) = 10^{2^x} \).

Key Concepts

Function Domain

The domain of a function is the set of all possible input values (usually represented by a variable like \( x \)), for which the function is defined. In other words, it's where the function "lives" on the number line. Understanding the domain is crucial when working with any kind of function, as it tells us the limitations and possible input values we can use without causing mathematical hiccups, like dividing by zero or taking the logarithm of a negative number.

In problems involving logarithmic functions, such as \( f(x)=\log _{2}(\log_{10} x) \), determining the domain involves ensuring that each function inside doesn't receive input that results in undefined outputs. For instance, with \( \log_{10} x \), we need \( x > 0 \) because a logarithm cannot take a non-positive number as an input. Then, taking \( \log_2(y) \) where \( y = \log_{10} x \), means \( y \) must also be greater than zero, which translates to \( x > 1 \). This makes the domain of the entire function \( x > 1 \).

Always remember:

• Check every nested function.
• Combine conditions to find the most restrictive domain.

Logarithmic Functions

Logarithmic functions are the opposite of exponential functions and are very important in the study of mathematical functions. A logarithm answers the question: to what exponent must a base be raised, to produce a given number? The general form is \( \log_b(a) = c \), which simply means that \( b^c = a \).

Logarithms come in various bases, but the most common are base 10 (common logarithms, \( \log(x) \)) and base \( e \) (natural logarithms, \( \ln(x) \)). In our function \( f(x) = \log _{2}(\log _{10} x) \), you’ve got a double layer of logs. The inner function \( \log _{10} x \) requires \( x > 0 \). The outer function takes it further to \( \log_2(a) = b \), meaning \( 2^b = a \). 

With logarithmic functions, it’s important to:

• Ensure the arguments are positive for real number results.
• Understand the number of layers: here we have an outer log applied to an inner log output.

Composition of Functions

Composition of functions involves creating a new function by applying one function to the result of another. If you have two functions, say \( f(x)\) and \( g(x)\), the composition \( (f \circ g)(x) \) means you plug the output of \( g(x) \) into \( f(x) \).

In the given example, \( f(x) = \log_2(\log_{10} x) \), we have a composition where the output of \( \log_{10} x \) becomes the input for \( \log_2 \). Understanding composition can simplify complex functions, especially when dealing with inverses.

Remember when composing functions:

• Check domains of both individual and composed functions.
• Evaluate the inner function first as it progresses the input through a chain.