Problem 20 - Thomas Calculus

Question

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\) $$ f(x)=x^{4}, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[4]{x} \) with domain and range both \( x \geq 0 \).

Step 1: Understand the Function

The given function is \( f(x) = x^4 \) for \( x \geq 0 \). This means the function maps non-negative values of \( x \) to their fourth power.

Step 2: Find the Inverse Function

To find the inverse, we need to express \( x \) in terms of \( y \) by solving \( y = x^4 \) for \( x \). Taking the fourth root of both sides yields \( x = \sqrt[4]{y} \). So, \( f^{-1}(x) = \sqrt[4]{x} \).

Step 3: Determine the Domain and Range of the Inverse Function

Since the original function is defined for \( x \geq 0 \), its output \( y \) is also non-negative. Therefore, the domain of \( f^{-1}(x) \) is \( x \geq 0 \). Similarly, the range of \( f^{-1}(x) \) is \( y \geq 0 \), as taking the fourth root will always yield a non-negative result.

Step 4: Verify the Inverse Function

To verify, we check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) = f(\sqrt[4]{x}) = (\sqrt[4]{x})^4 = x \) holds true. Similarly, \( f^{-1}(f(x)) = f^{-1}(x^4) = \sqrt[4]{x^4} = x \) for \( x \geq 0 \), also holds true.

Key Concepts

Domain and Range

In the world of functions, understanding the domain and range is crucial. The domain refers to all possible input values that a function can accept. In this example, the function is \( f(x) = x^4 \) with \( x \geq 0 \). This means that the domain is all non-negative real numbers because any negative number raised to the fourth power wouldn't make sense in this context.

The range, on the other hand, is all possible output values a function can produce. For \( f(x) = x^4 \), the output is also non-negative. Thus, the range for this function consists of all non-negative real numbers. Essentially, both the domain and range of this original function are the same: \( \{ x \ | \ x \geq 0 \} \).

When you find the inverse function, its domain and range are swapped compared to the original function. So for \( f^{-1}(x) = \sqrt[4]{x} \), the domain becomes what was the range of \( f(x) \), and the range turns into what was the domain of \( f(x) \). This ensures that all calculations work correctly, keeping the consistency across transformations.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. This is a vital process in verifying inverse functions. For a function \( f(x) \) and its inverse \( f^{-1}(x) \), verifying involves showing that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Let's break down the process:

• First, compute \( f(f^{-1}(x)) = f(\sqrt[4]{x}) \). The function \( f(x) \) maps \( \sqrt[4]{x} \) to its fourth power. Thus, \( (\sqrt[4]{x})^4 = x \).
• Next, calculate \( f^{-1}(f(x)) = f^{-1}(x^4) \). The inverse function \( f^{-1}(x) \) maps \( x^4 \) back to \( x \) by taking the fourth root: \( \sqrt[4]{x^4} = x \).

This symmetry confirms the relationship between a function and its inverse. Understanding composition helps in validating the accuracy and correctness of inverses, ensuring mathematical concepts stay intact.

Fourth Root

The fourth root is a unique mathematical operation. It is the process of finding a number's fourth root is one of reversing the effect of raising a number to the fourth power.

When you see \( \sqrt[4]{x} \), you're looking for a number which, when raised to the fourth power, equals \( x \). Essentially:

• The fourth root of 16 is 2 because \( 2^4 = 16 \).
• Likewise, the fourth root of 81 is 3, since \( 3^4 = 81 \).

In our problem, finding \( f^{-1}(x) = \sqrt[4]{x} \) means that for every \( x \) that is non-negative, the function returns its fourth root. This operation only works for non-negative numbers in this context since our original function \( f(x) = x^4 \) was limited to \( x \geq 0 \).

Understanding roots, especially higher roots, expands one's ability to reverse exponential operations, a foundational skill in algebra and beyond.