Problem 42 - Calculus of a Single Variable

Question

Find the inverse function of \(f,(\mathbf{b})\) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, ( \(\mathbf{c} )\) describe the relationship between the graphs, and ( \(\mathbf{d} )\) state the domain and range of \(f\) and \(f^{-1} .\) \(f(x)=\sqrt{x^{2}-4}, \quad x \geq 2\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \sqrt{x^{2}-4}\) is \(f^{-1}(x) = \sqrt{x^2 + 4}\). Its graph is a reflection of \(f(x)\) over the line \(y = x\). The domain of \(f(x)\) is \([2, \infty)\) and its range is \([0, \infty)\). Similarly, the domain of \(f^{-1}(x)\) is \([-2, \infty)\) and its range is \([0, \infty)\).

Step 1: Find the Inverse Function

To find the inverse of \(f(x) = \sqrt{x^{2}-4}\), you exchange \(y\) and \(x\) and then, solve for \(y\). Thus, \(y = \sqrt{x^{2}-4}\) would be changed to \(x = \sqrt{y^{2} - 4}\). Squaring both sides, \(x^2 = y^2 - 4\) leads to \(y^{2} = x^{2} + 4\). This means \(y = \sqrt{x^{2} + 4}\) or \(y = -\sqrt{x^2 + 4}\), however since \(x \geq 2\) originally, the inverse function is \(f^{-1}(x) = \sqrt{x^2 + 4}\) as the positive square root is enough.

Step 2: Graph the Functions

Sketch the graph of \(f(x) = \sqrt{x^{2}-4}\) and it's inverse \(f^{-1}(x) = \sqrt{x^2 + 4}\) on the same set of axes. Remember, the graph of \(f(x)\) is limited to \(x \geq 2\) and the graph of \(f^{-1}(x)\) will always be positive due to the square root.

Step 3: Describe the relationship between the graphs

The graph of \(f^{-1}(x)\) will be a reflection of the graph of \(f(x)\) over the \(y = x\) line. In other words, for every point on \(f(x)\) that has coordinates (a, b), the point on \(f^{-1}(x)\) will have coordinates (b, a). This is a general property of functions and their inverses.

Step 4: State the Domain and Range

For the function \(f(x) = \sqrt{x^{2}-4}\), the domain is \([2, \infty)\), and the range is \([0, \infty)\). After identifying the inverse function \(f^{-1}(x) = \sqrt{x^2 + 4}\), its domain is \([-2, \infty)\), and its range is \([0, \infty)\].

Key Concepts

Graphing Functions

Graphing functions and their inverses helps visualize their relationships. When you're graphing a function like \( f(x) = \sqrt{x^2 - 4} \), you'll want to focus on just the portion where \( x \geq 2 \). The curve starts at the point \( (2, 0) \) and rises sharply as \( x \) increases. The graph of its inverse, \( f^{-1}(x) = \sqrt{x^2 + 4} \), can be plotted on the same set of axes. This graph will always be in the positive quadrant because the square root function only produces non-negative results.When looking at the graphs, remember they should mirror each other over the line \( y = x \). Visualizing this symmetry helps in understanding how the inverse functions reflect each other, enhancing comprehension of their dual nature. Keep this trick in mind: If you fold the graph paper along the line \( y = x \), the graphs of \( f(x) \) and \( f^{-1}(x) \) should align if they are inverses.

Function Domain

The domain of a function describes all the possible input values (usually \( x \)) that the function can accept. For the function \( f(x) = \sqrt{x^2 - 4} \), the domain is restricted to ensure that the expression inside the square root is non-negative. Here, \( x \geq 2 \) ensures that \( x^2 - 4 \geq 0 \). Therefore, the domain of \( f(x) \) is \([2, \infty)\).Understanding a function's domain is crucial. It determines where the function "lives" on the graph, indicating the valid x-values you can plug into the function. For the inverse function \( f^{-1}(x) = \sqrt{x^2 + 4} \), the domain isn't limited by non-negativity because any real number put into \( x^2 + 4 \) will be positive. Thus, the domain for \( f^{-1}(x) \) is \([-2, \infty)\), showing its broad range of allowable inputs.

Function Range

The range of a function specifies all the possible output values (usually \( y \)), the results after applying the function to its domain. For \( f(x) = \sqrt{x^2 - 4} \), since \( x \geq 2 \), the results of the function are non-negative values starting from zero upwards. Thus, the range of \( f(x) \) is \([0, \infty)\).For the inverse function \( f^{-1}(x) = \sqrt{x^2 + 4} \), the situation is a bit different. Even though this expression always produces a number greater than the square root of 4, because of the definition of inverse functions aligning their range with the domain of the original function, the range remains \([0, \infty)\).Understanding the range helps us know how high and low the graph of a function can go. Exploring ranges gives insight into the behavior and potential outcomes of applying the function across its entire span of operations.

Relations of Functions and Their Inverses

The relation between functions and their inverses is characterized by symmetry over the line \( y = x \). This means that the graphs of \( f(x) \) and \( f^{-1}(x) \) are mirror images across this line.For instance, in our exercise, any point \( (a, b) \) on \( f(x) = \sqrt{x^2 - 4} \) will have a corresponding point \( (b, a) \) on the inverse \( f^{-1}(x) = \sqrt{x^2 + 4} \). This reflects a fundamental property: applying a function and then its inverse returns the original value.View it this way:

• If you have a function \( f \), then the operation \( f(f^{-1}(x)) = x \) holds true.
• Similarly, applying the inverse after the function, \( f^{-1}(f(x)) = x \), yields \( x \) back.

Grasping these relationships makes it easier to understand how inverse functions work in mathematics, effectively reversing any produced result from the original function back to the starting value.