Problem 40 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-x^{3}+4$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = -x^{3} + 4\) is \(f^{-1}(x) = \sqrt[3]{4 - x}\). When graphed together on the same axes, these two functions reflect about the line \(y=x\).

Step 1: Find the Inverse Function

Switch \(x\) and \(y\), so the original function \(f(x) = -x^{3} + 4\) becomes \(x = -y^{3} + 4\). Solve for \(y\), as you're attempting to find the inverse function \(f^{-1}(x)\). Firstly, subtract 4 from both sides, yielding \(x - 4 = -y^{3}\). Multiply by -1 to get \(4 - x = y^{3}\). Lastly, to isolate \(y\), take the cube root from both sides which gives \(f^{-1}(x) = \sqrt[3]{4 - x}\).

Step 2: Graph the Original Function

To graph the function \(f(x) = -x^{3} + 4\), choose some values for x and compute the corresponding y-values by plugging into the function. For example, select values like -2, -1, 0, 1, 2. Graph these points and draw a curve through them, making sure to indicate the direction the graph extends for values beyond those graphed.

Step 3: Graph the Inverse Function

Similarly, to graph the inverse function \(f^{-1}(x) = \sqrt[3]{4 - x}\), you will again choose some x-values and calculate the corresponding y-values by plugging into the inverse function. After graphing these points, draw the curve. Since this function is the inverse of the original, reflect the original graph on the line \(y=x\).

Step 4: Graph Reflection Line

Lastly, to visually convey the relationship between the function and its inverse, graph the line \(y=x\) to illustrate that the inverse function is indeed a reflection of the original across that line.

Key Concepts

Graphing Functions

When graphing a function such as \(f(x) = -x^3 + 4\), it is essential to understand how the graph represents the relationship between input values (\(x\)) and output values (\(f(x)\)). A graph can be constructed by choosing a variety of \(x\) values, often around the point where the function changes direction, known as a turning point or vertex. For cubic functions, plotting points symmetrically around the origin can help reveal the graph's nature.

Once you have calculated the corresponding \(f(x)\) values, plot these points on a coordinate system. By connecting these points smoothly, you illustrate the continuous nature of the function. Note that for real world applications, understanding the behavior of a function beyond the plotted points and towards infinity is vital. The end behavior of the function \(f(x) = -x^3 + 4\) tells us that as \(x\) approaches positive and negative infinity, the function will extend downwards in both directions.

Cube Root

The cube root, denoted as \(\root{3}{x}\), is the inverse operation of cubing a number. For a given number \(y\), if \(y = x^3\), then \(x\) is the cube root of \(y\). In the context of finding the inverse function, we apply the cube root to both sides of an equation to isolate the variable.

For instance, if we have \(4 - x = y^3\), by taking the cube root of both sides, we find \(y = \root{3}{4 - x}\), which represents the inverse function. Understanding the cube root is crucial not only for solving equations but also for graphing functions that involve this operation, as it dramatically affects the shape of the graph.

Function Transformation

Function transformation involves altering the basic graph of a function in various ways—such as shifting, stretching, compressing, or reflecting—to produce a new graph.

These transformations can be categorized as:

• Horizontal and vertical translations, which shift the graph left or right and up or down respectively,
• Horizontal and vertical stretches/compressions, which elongate or shrink the graph in respective directions,
• Reflections across the \(x\)-axis or \(y\)-axis, changing the orientation of the graph.

The process of finding the inverse of a function often involves reflecting its graph across the line \(y=x\), transforming each point \((x, y)\) to \((y, x)\). This is a distinctive type of transformation since it alters both the horizontal and vertical coordinates concurrently.

Reflection Across the Line y=x

Reflection across the line \(y=x\) is a specific type of function transformation that interchanges the input and output of a function. This operation is fundamental when finding the inverse of a function.

To understand this concept visually, imagine the line \(y=x\) as a mirror. Each point of the original function is 'reflected' over this line to obtain the graph of the inverse function. For every point \((a, b)\) on the original function, there will be a corresponding point \((b, a)\) on the inverse function.

Graphically, this line of reflection provides a symmetry that helps to validate whether a function is correctly inverted. If the function and its inverse are accurate mirrors of each other across the line, then they are perfectly reflected, proving the inverse has been found correctly. Remember that a function is invertible if and only if it passes the horizontal line test, meaning each horizontal line intersects the graph at most once.