Problem 63 - College Algebra

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and \(g\) are inverses. $$f(x)=\sqrt[3]{x}-2, g(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

By comparing the graphs of \(f(x)\) and \(g(x)\) with each other and their reflection across the line \(y=x\), it can be visually determined that they are indeed inverses of each other.

Step 1: Graph function \(f\)

Using a graphing utility, plot \(f(x) = \sqrt[3]{x} - 2\). Remember to plot over a sensible domain, like \(-10 \leq x \leq 10\), for a versatile view of the function.

Step 2: Graph function \(g\)

Now in the same viewing window plot the function \(g(x) = (x+2)^3\). Pay attention to how this graph relates to the graph of function \(f\).

Step 3: Graph the line \(y=x\)

After plotting both functions, next plot the line \(y=x\). This line will act as a mirror and if functions \(f\) and \(g\) are inverses, their graphs should mirror each other across this line.

Step 4: Visually check for inverses

Now check if \(f\) and \(g\) are inverses visually by comparing their shapes and positions relative to each other and the line \(y=x\). If \(f\) and \(g\) are inverses, they should reflect each other across \(y=x\).

Key Concepts

Graphing Utility

When dealing with complex algebraic functions, a graphing utility is an invaluable tool that can visually display relationships between variables. A graphing utility allows you to enter the equation of a function and produces a graph based on the values you input. It's perfect for exploring how changing the equation changes the graph, and for verifying theoretical predictions like the relationship between a function and its inverse.

For example, in the exercise provided, the functions are plotted in a shared viewing rectangle, which permits a clear comparison and helps in identifying whether the functions are inverses of each other. The section on inverse functions will expand on why this visualization is important.

Inverse Functions

Inverse functions are like two sides of the same coin. For a given function, its inverse essentially reverses the process. If you think of a function as a machine where you input 'x' and get 'y', the inverse function takes 'y' and gives you back 'x'.

To determine if two functions are inverses, one method is to graph them both along with the line y = x. If they're true inverses, they'll reflect over this line. So if you folded the graph along y = x, one function would land right on top of the other. The exercise demonstrates this graphically, allowing students to see the symmetry visually, which links back to the graphing utility's value in understanding complex concepts.

Cubic Root Function

The cubic root function, represented as f(x) = \( \sqrt[3]{x} \) in the exercise, is an example of an inverse function. Just as squaring a number and taking the square root are inverse operations, cubing a number and taking the cubic root are inverses too. A cubic root function undoes the action of a cubic function.

This inverse relation is why a cubic root function graph will mirror a cubic function graph across the line y = x. When you're plotting the cubic root function using a graphing utility, you'll notice that its shape is distinct, somewhat resembling the letter S leaning forwards, and will occupy both the positive and negative side of the y-axis, reflecting its ability to accept both positive and negative inputs.

Cubic Function

On the other hand, the cubic function in the exercise, g(x) = (x + 2)^3, shows how a simple algebraic function can be transformed and then graphed. This function represents an algebraic equation where the variable x is subject to cubing, after an initial addition of 2.

When graphed on a utility, the cubic function's curve has a distinct shape, rising and falling steeply, and stretching out to infinity on both ends. This particular function will also shift units to the left on the graph due to the '+2' inside the cubing operation. If you were to draw a line from each point on the cubic function to the corresponding point on the cubic root function, these lines would meet at right angles - an elegant proof of the relationship between a function and its inverse.