Problem 41 - 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 \(f^{-1} .\) \(f(x)=\sqrt{4-x^{2}}, \quad 0 \leq x \leq 2\)

Step-by-Step Solution

Answer

The inverse function of \(f(x)=\sqrt{4-x^{2}}\) is \(f^{-1}(x) = -\sqrt{4-x^{2}}\). Both functions, when graphed, are semicircles with a radius of 2 centred at the origin - with \(f\) in the upper plane and \(f^{-1}\) in the lower plane. They are mirror images along the line \(y=x\). The domain and range of \(f(x)\) are \([0,2]\) and \([0,2]\) respectively, while the domain and range of \(f^{-1}(x)\) are \([0,2]\) and \([-2,0]\) respectively.

Step 1: Find the inverse function

To find the inverse function of \(f(x)=\sqrt{4-x^{2}}\), first replace \(f(x)\) with \(y\): \(y=\sqrt{4-x^{2}}\). Next, switch the roles of \(x\) and \(y\) to get: \(x=\sqrt{4-y^{2}}\). Then, square both sides to remove the square root, resulting in: \(x^{2} = 4 - y^{2}\). Solving for \(y\), we have: \(y^{2} = 4 - x^{2}\) or \(y = \sqrt{4 - x^{2}}\) and \(y = -\sqrt{4 - x^{2}}\). From the given range of the original function, we see that the inverse is \(y = -\sqrt{4 - x^{2}}\).

Step 2: Graph \(f(x)\) and \(f^{-1}(x)\)

To graph \(f(x)=\sqrt{4-x^{2}}\) and \(f^{-1}(x) = -\sqrt{4 - x^{2}}\), plot each function on the same set of coordinates. \(f(x)\) is a semicircle with radius 2 centred at the origin, lying in the upper plane (due to \(0 \leq x \leq 2\)), while \(f^{-1}(x)\) is a semicircle with radius 2 centred at the origin, located in the lower plane.

Step 3: Describe the relationship between the graphs

Observing the graphs of \(f(x)\) and \(f^{-1}(x)\), we see that they are mirror images of each other with respect to the line \(y=x\). This is a property that applies to all pairs of functions and their inverses.

Step 4: State the domain and range of \(f(x)\) and \(f^{-1}(x)\)

The domain of \(f(x)\) is the set of all values of \(x\) for which \(f(x)\) is defined. Here, \(f(x) = \sqrt{4-x^{2}}\) is defined for \(0 \leq x \leq 2\), meaning the domain of \(f(x)\) is \([0,2]\). The range of \(f(x)\) is the set of all possible outputs or y-values, which is here \(0 \leq y \leq 2\), so the range of \(f(x)\) is \([0,2]\). For the inverse function \(f^{-1}(x) = -\sqrt{4 - x^{2}}\), given that it's the mirror image of \(f(x)\) along \(y=x\), we can conclude that the domain of \(f^{-1}(x)\) is \([0,2]\) and the range of \(f^{-1}(x)\) is \([-2,0]\).

Key Concepts

Domain and Range

The domain and range of a function play a crucial role in understanding and describing the behavior of functions and their inverses. The domain of a function, like the original function \( f(x) = \sqrt{4-x^2} \), includes all allowable x-values for which the function is defined. For this particular function, since it involves a square root, it's defined for \( 0 \leq x \leq 2 \). The domain thus is \([0,2]\). Furthermore, the range of \( f(x) \) is determined by the possible values that result from the function, which are y-values spanning from 0 to 2, and hence, the range is also \([0,2]\).

• Domain of \( f(x) \): \([0,2]\)
• Range of \( f(x) \): \([0,2]\)

Now, to find the inverse, \( f^{-1}(x) = -\sqrt{4-x^2} \), we interchange the domain and range. Here, the domain becomes \([0,2]\), while the range adjusts to \([-2,0]\) since the function represents the lower semicircle of the same radius. Understanding how domains and ranges swap for inverse functions helps in verifying the properties and relationships between functions and their inverses.

• Domain of \( f^{-1}(x) \): \([0,2]\)
• Range of \( f^{-1}(x) \): \([-2,0]\)

Graphing Functions

One of the most informative ways to analyze a function and its inverse is through graphing. When graphed, the function \( f(x) = \sqrt{4-x^2} \) appears as a semicircle in the upper half of the coordinate plane. It covers an arc from \((2,0)\) to \((0,2)\) and crosses back through to \((-2,0)\), encapsulating the domains and ranges available.
The inverse function, \( f^{-1}(x) = -\sqrt{4-x^2} \), finds its place in the lower semicircle of the coordinate plane. This function's arc moves along \((2,0)\) to \((0,-2)\) and again to \((-2,0)\), reciprocating the transformations of \( f(x) \).

• Both semicircles maintain symmetry around the y-axis, emphasizing inherent symmetry properties.
• They highlight their inversion about the line \( y = x \), marking a graphically intuitive transformation.

Graphing functions and their inverses not only provides visual insight but also confirms calculated domains and ranges.

Function Properties

When exploring function properties, several characteristics stand out, particularly for functions such as \( f(x) = \sqrt{4-x^2} \) and its inverse. Vital properties include:

• Symmetry: The function and its inverse show symmetric properties around the line \( y = x \). This is typical for inverse functions, where the graph of one function typically reflects across the line \( y = x \) to form the graph of the other.
• Restricting on Domains: Square root functions often require domain restrictions to remain real and defined. Understanding why the range is limited from 0 to 2 in both original and inverse forms involves recognizing restrictions placed due to the square root function and the non-negativity it inherently demands.
• Continuity: Both functions are continuous within their respective domains, meaning there are no breaks or gaps in the arcs of the semicircles as plotted on the graph.

Exploring these properties gives a deeper understanding of how function transformations can affect the graphs, allowing students to grasp how reflections, domains, and ranges contribute to defining inverse relationships.