Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=\frac{6}{\sqrt{x}}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=6/\sqrt{x}\) is \(f^{-1}(x)=36/x^2\). Both functions, when graphed, are reflections of each other across the line \(y = x\).

Step 1: Swap x and y

To find the inverse function, we first replace \(f(x)\) with \(y\), giving \(y=6/\sqrt{x}\). Then we swap x and y to get \(x = 6/\sqrt{y}\).

Step 2: Solve for y

Multiply each side by \(\sqrt{y}\) to remove y from the denominator: \(x\sqrt{y} = 6\). Then, to isolate y, square both sides. Squaring \(x\sqrt{y}\) gives \(x^2y\) while squaring 6 gives 36. So, \(x^2y = 36\). Finally, dividing both sides by \(x^2\) will provide the equation of the inverse function: \(y=36/x^2\). Therefore, \(f^{-1}(x)=36/x^2\).

Step 3: Graph f(x) and \(f^{-1}(x)\)

Using a graphing utility, both \(f(x)\) and \(f^{-1}(x)\) should be graphed. The function \(f(x)=6/\sqrt{x}\) should exhibit a decrease as x increases. In contrast, the inverse function \(f^{-1}(x)=36/x^2\) should increase as x decreases, resulting in the graphs reflected over the line \(y=x\).

Step 4: Describe the relationship between the graphs

Observing the graph, we should see that these two functions are reflections of each other across the line \(y=x\). This is a defining characteristic of inverse functions, and it confirms correct calculation of the inverse.

Key Concepts

Graphing Inverse Functions

To visualize and better understand the relationship between a function and its inverse, graphing is an essential tool. For the given exercise, the function is defined as \(f(x)=\frac{6}{\sqrt{x}}\). When graphing its inverse, the first step involves swapping the variables: we replace \(f(x)\) with \(y\) and then interchange \(x\) and \(y\) to find an equation that solves for \(y\).

Once the inverse function \(f^{-1}(x)\) is algebraically determined, graphing both functions on the same set of axes is key. Typically, a graphing utility or software is used to do this efficiently. The function \(f(x)\) and its inverse \(f^{-1}(x)\) will be reflections of one another over the line \(y=x\). This is because for every point \((a, b)\) on the graph of \(f(x)\), there will be a corresponding point \((b, a)\) on the graph of \(f^{-1}(x)\).

To ensure clarity and precision while graphing, it's important to consider the scale and the range of values taken by \(x\) and \(y\). For example, since the original exercise involves a square root function, we must remember that \(x\) must be positive for \(f(x)\). Similarly, the inverse will only be valid for positive \(x\) values resulting from \(f^{-1}(x)\).

Moreover, seeing the two graphs intersect the line \(y=x\) at the point where \(f(x) = x\) and \(f^{-1}(x) = x\) helps reinforce the concept that an inverse function 'undoes' the action of the original function. This visualization aids in deeper understanding of the idea that the inverse of a function essentially switches the roles of inputs and outputs.

Solving Inverse Functions Algebraically

Algebraically solving for the inverse of a function may seem challenging, but following systematic steps can simplify the process. Using the provided exercise \(f(x)=\frac{6}{\sqrt{x}}\) as a reference, let's elaborate on these steps in detail.

Firstly, symbolize the function as \(y\) instead of \(f(x)\), creating the equation \(y=\frac{6}{\sqrt{x}}\). The goal then is to solve for \(x\) as a function of \(y\). This inversion process includes interchanging \(x\) and \(y\) to get \(x = 6/\sqrt{y}\).

The next phase involves manipulating this new equation to isolate \(y\). Multiplying each side of the equation by \(\sqrt{y}\) and squaring both sides to remove the square root leads to \(x^2y = 36\). Finally, divide by \(x^2\) to solve for \(y\), yielding the inverse function \(f^{-1}(x)=36/x^2\).

Some tips to ensure a solid understanding: keeping track of each algebraic manipulation is crucial to avoid errors, especially when dealing with square roots and other operations that affect the entire equation. Additionally, it's important to check the results by plugging in values or by confirming graphically that the obtained inverse satisfies the relationship of being a reflection over the line \(y=x\).

Properties of Inverse Functions

Inverse functions are grounded in the concept of reversibility. For any function \(f(x)\), its inverse \(f^{-1}(x)\) effectively undoes whatever operation \(f\) does. This characteristic leads to key properties that all inverse functions share.

Firstly, if a function \(f\) is defined such that every output corresponds to one and only one input, then \(f\) is said to be 'one-to-one' or 'injective.' This property is vital because it guarantees that the inverse function \(f^{-1}\) will also be a function, meaning every input will produce only one output.

Furthermore, for each pair of a function and its inverse, applying one after the other will result in the original value. This is expressed mathematically as: \[f^{-1}(f(x)) = x\] and \[f(f^{-1}(x)) = x\].

Another important property is symmetry: the graph of a function and its inverse are mirror images along the line \(y=x\). It's interesting to note that the roles of the domain and range are swapped in a function and its inverse. That is, the domain of \(f\) becomes the range of \(f^{-1}\) and vice versa.

These properties can be used not only as a verification tool to check if the solution to an inverse function is correct but also to gain insights into the behavior of functions and their inverses in various mathematical contexts. Understanding these properties is foundational for higher learning in algebra and calculus, where inverse functions play a significant role.