Problem 71 - A Graphical Approach to Precalculus with Limits

Question

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x^{2}-4}, x \geq 2$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x^2 + 4} \).

Step 1: Understand the Function Definition

The function given is \( f(x) = \sqrt{x^2 - 4} \) with \( x \geq 2 \). This function is defined for all values of \( x \) such that \( x^2 - 4 \geq 0 \), i.e., \( x \geq 2 \), meaning it is a one-to-one function as specified.

Step 2: Express y in terms of x

To find the inverse, first express the function in terms of \( y \, = \, f(x) \):\[ y = \sqrt{x^2 - 4} \]

Step 3: Square Both Sides

Eliminate the square root by squaring both sides:\[ y^2 = x^2 - 4 \]

Step 4: Solve for x

Isolate \( x^2 \) by adding 4 to both sides:\[ x^2 = y^2 + 4 \]

Step 5: Solve for x with the correct conditions

Since \(x \geq 2\), when taking the square root, we consider the positive root:\[ x = \sqrt{y^2 + 4} \]

Step 6: Write the Inverse Function

Express \( x \) in terms of \( y \), considering it as the inverse function:\[ f^{-1}(x) = \sqrt{x^2 + 4} \] where \( x \geq 0 \) as derived from squaring.

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each input corresponds to a unique output, and each output corresponds to a unique input. Understanding if a function is one-to-one is important because it helps us determine if an inverse function exists. A simple way to check if a function is one-to-one is by using the Horizontal Line Test.

• Draw a horizontal line across the graph of the function.
• If any horizontal line intersects the graph more than once, the function is not one-to-one.
• If no horizontal line intersects the graph more than once, the function is one-to-one.

In our context with the function \( f(x) = \sqrt{x^2 - 4} \), we can understand that it's one-to-one by recognizing the behavior of square root functions which are typically one-to-one when restricted properly. When expressed in terms of \( x \geq 2 \), it ensures the function remains one-to-one.

Domain and Range

Understanding the domain and range of a function is essential in analyzing how the function behaves. The domain is the set of all possible input values (\( x \)) for the function, while the range is the set of all possible output values (\( y \)). For the function \( f(x) = \sqrt{x^2 - 4} \), determining the domain involves ensuring the expression inside the square root is non-negative.

• The function is defined when \( x^2 - 4 \geq 0 \), which simplifies to \( x \geq 2 \).
• Therefore, the domain of \( f(x) \) is \([2, \infty)\).

When considering the range, we look at the values \( f(x) \) can take.

• Since the square root function produces non-negative outputs, the smallest value \( f(x) \) can be is 0 when \( x = 2 \).
• The range of \( f(x) \) thus becomes \([0, \infty)\).

Knowing the domain and range helps in finding the inverse, ensuring that the inverse function is correctly defined and interpretable.

Square Root Functions

Square root functions are functions that involve the square root of a variable or expression. In the function \( f(x) = \sqrt{x^2 - 4} \), the square root affects how the function behaves visually and mathematically.Square root functions typically:

• Start from a certain value and increase slowly because squaring grows quite fast.
• Always produce non-negative outputs by default.

The behavior of square root functions is influenced by their domain restrictions. In our example, the restriction \( x \geq 2 \) ensures that the expression inside the square root stays non-negative. This restriction also helps define the inverse function correctly by ensuring all terms are properly defined.Another interesting property of square root functions is their impact on inverse functions. When computing the inverse of a function involving a square root, it's critical to consider conditions such as non-negativity.These properties dictate how you maneuver algebraic manipulation when working on problems involving inverses of square root functions.