Problem 17 - Thomas Calculus

Question

Each of Exercises \(13-18\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1}\) . Find a formula for \(f^{-1}\) in each case. $$ f(x)=(x+1)^{2}, \quad x \geq-1 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x} - 1 \), for \( x \geq 0 \).

Step 1: Understanding the Function and Its Inverse

We are given the function \( f(x) = (x+1)^2 \) with the condition \( x \geq -1 \). Our task is to find the inverse function \( f^{-1}(x) \) by reversing the roles of \( x \) and \( y \).

Step 2: Interchanging Variables

Replace \( f(x) \) with \( y \), so we have \( y = (x+1)^2 \). To find \( f^{-1}(x) \), we interchange \( x \) and \( y \), leading to \( x = (y+1)^2 \).

Step 3: Solving for the Inverse

Take the square root of both sides to solve for \( y \) in terms of \( x \): \( \sqrt{x} = y + 1 \). Hence, \( y = \sqrt{x} - 1 \).

Step 4: Addressing the Domain of the Inverse

Since the original domain of \( x \) in \( f(x) \) is \( x \geq -1 \), \( f(x) = y \) produces outputs \( y \geq 0 \). Therefore, the domain for \( f^{-1}(x) \) is \( x \geq 0 \), conforming to \( y = \sqrt{x} - 1 \).

Step 5: Stating the Formula for the Inverse Function

The function \( f^{-1}(x) \) is given by \( f^{-1}(x) = \sqrt{x} - 1 \) where \( x \geq 0 \).

Key Concepts

Solving Equations

Finding the inverse of a function involves a few straightforward steps that rely on solving equations. The main goal is to interchange based on the variables given in the function equation. Initially, we have a function given as \( f(x) = (x+1)^2 \) with a restriction of \( x \geq -1 \).

• Start by letting \( y = f(x) \), giving us \( y = (x+1)^2 \).
• To find the inverse, we swap \( x \) and \( y \), converting the equation into \( x = (y+1)^2 \).
• The next step is to isolate \( y \). This is achieved by taking the square root of both sides: \( \sqrt{x} = y + 1 \).
• Finally, solve for \( y \): \( y = \sqrt{x} - 1 \).

This final equation gives the expression we need for the inverse function, \( f^{-1}(x) = \sqrt{x} - 1 \), allowing us to understand the inverted relationship between variables.

Function Graphs

Graphs serve as a powerful visual aid in understanding inverse functions. For the original function \( f(x) = (x+1)^2 \), which is a parabola, its graph will start at \( x = -1 \) and open upwards. The vertex of this parabola is at \( (-1, 0) \).

• In contrast, the inverse function \( f^{-1}(x) = \sqrt{x} - 1 \) will graphically appear quite different. This function is a translation of the basic square root function.
• It starts from \( x = 0 \) and the output begins at \( y = -1 \), tracing a horizontal shape similar to the upper branch of a sideways parabola.
• The graph of \( f^{-1}(x) \) gives a clear reflection of \( f(x) \) over the line \( y = x \), showcasing the switch in roles of input and output.

Understanding these graphs helps you see the symmetry and relation between a function and its inverse, providing an intuitive grasp alongside the algebraic process.

Domain and Range

The domain and range of a function and its inverse are crucial in defining their respective behaviors. For \( f(x) = (x+1)^2 \), the domain is restricted to \( x \geq -1 \).

• The range of \( f(x) \) is derived from its output values, leading to \( y \geq 0 \).
• When it comes to the inverse function \( f^{-1}(x) = \sqrt{x} - 1 \), the roles are switched. The domain -- i.e., accepted input values -- becomes \( x \geq 0 \), as these reflect the original range of \( f(x) \).
• Consequently, the range of the inverse is \( y \geq -1 \), which is determined by substituting the smallest allowable input, \( x = 0 \), into \( \sqrt{x} - 1 \).

Recognizing how domain and range transform between a function and its inverse enables you to fulfill crucial algebraic conditions and ensures no values fall outside the defined scope, preserving mathematical function integrity.