Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=(x-3)^{2}, x \geq 3 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x} + 3 \). Verification shows \( f^{-1}(f(x))=x \) and \( f(f^{-1}(x))=x \).

Step 1: Understand the Function

The given function is \( f(x) = (x-3)^2 \) with the condition that \( x \geq 3 \). This restriction ensures that the function is one-to-one and thus invertible on this domain.

Step 2: Solve for the Inverse Function

To find the inverse function \( f^{-1}(x) \), start by setting \( y = (x-3)^2 \). Solve for \( x \) in terms of \( y \):\[(x-3)^2 = y\]Take the square root of both sides: \( x - 3 = \sqrt{y} \). Since \( x \geq 3 \), we only consider the non-negative square root, yielding \( x = \sqrt{y} + 3 \). Thus, the inverse function is \( f^{-1}(x) = \sqrt{x} + 3 \).

Step 3: Verify \(f^{-1}(f(x))=x\)

Let \( f^{-1}(x) = \sqrt{x} + 3 \) and substitute \( f(x) \) into it: \[f^{-1}(f(x)) = f^{-1}((x-3)^2) = \sqrt{(x-3)^2} + 3\]. Since \( x \geq 3 \) and square root and square are inverse operations, \( \sqrt{(x-3)^2} = x-3 \). Adding 3 back gives \( f^{-1}(f(x)) = x \).

Step 4: Verify \(f(f^{-1}(x))=x\)

Substitute \( f^{-1}(x) = \sqrt{x} + 3 \) into \( f(x) \): \[f(f^{-1}(x)) = f(\sqrt{x} + 3) = ((\sqrt{x} + 3)-3)^2 = (\sqrt{x})^2 = x \]. Thus, \( f(f^{-1}(x)) = x \), completing the verification.

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, is a type of mathematical function where each element of the domain is paired with a unique element of the codomain. This uniqueness is crucial because it guarantees that no two different inputs will produce the same output.

In other words, a one-to-one function ensures that for any two inputs, say \(x_1\) and \(x_2\), if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). Without this property, it would not be possible to define a true inverse function.

• To confirm a function is one-to-one, check that it passes the **horizontal line test**: a horizontal line should intersect the graph of the function at most once.
• In the context of this exercise, the function \(f(x) = (x-3)^2\) is restricted to \(x \geq 3\), making it one-to-one within this domain.

This restriction ensures all outputs are distinct, thus allowing us to find its inverse.

Function Verification

Function verification involves demonstrating that two functions are truly inverses of each other. When a function \(f\) has an inverse \(f^{-1}\), two key relationships must hold: \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\). These relationships highlight the concept that applying a function and its inverse in succession should return the original value.

In our exercise, we verify these relationships by substitution and simplification:

• **Verification of \(f^{-1}(f(x)) = x\)**: Substitute \(f(x)\) into \(f^{-1}(x)\) and show that you return to the original \(x\).
• **Verification of \(f(f^{-1}(x)) = x\)**: Replace \(x\) with \(f^{-1}(x)\) in \(f(x)\) and simplify to get back to \(x\).

Both processes involve understanding and utilizing the properties of inverse operations, such as squaring and square rooting, to simplify back to \(x\). The verification confirms that our determined inverse is correct and satisfies the definition of an inverse function.

Square Root Function

A square root function involves finding a number that, when squared, yields another given number. In mathematical terms, for any non-negative number \(x\), the square root is signified as \(\sqrt{x}\).

The square root function is described by the equation \(y = \sqrt{x}\), which inherently involves positive values because we consider non-negative square roots. This function has a key property: reversing a squared value using a square root should retrieve the original non-negative value.

• The principal square root of a squared quantity \((x-3)^2\) is \(x-3\), as noted in the verification of \(f^{-1}(f(x)) = x\). The restriction of \(x \geq 3\) ensures that the square root reflects the non-negative domain.
• In an inverse scenario like our exercise, square roots undo the squaring operation, which is fundamental to solving the inverse and verifying its correctness.

Understanding the square root function is essential for operating with inverse functions, especially where square operations are involved, creating a fundamental balance between squaring and square rooting in mathematics.