Problem 39 - 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. $$f(x)=x^{3}-6$$

Step-by-Step Solution

Answer

The inverse function of the given function \(f(x) = x^{3} - 6\) is \(f^{-1}(x) = \sqrt[3]{x + 6}\). When graphed on the same set of axes, these two functions reflect each other about the line \(y=x\).

Step 1: Find the Equation of the Inverse

First, recall that to find the inverse of a function, we swap \(x\) and \(y\), and then solve for \(y\). To begin, let \(y = f(x) = x^3 - 6\). Swap \(x\) and \(y\) to get \(x = y^3 - 6\). Now, solve for \(y\) to find the inverse. Do this by isolating \(y\) on one side.

Step 2: Solve for the Inverse

To isolate \(y\), add 6 to both sides, and cube root both sides of the equation. Doing this yields \(y = \sqrt[3]{x + 6}\). Hence, the inverse function \(f^{-1}(x) = \sqrt[3]{x + 6}\)

Step 3: Graph Both Functions

Now, graph the function \(f(x) = x^3 - 6\) and the inverse function \(f^{-1}(x) = \sqrt[3]{x + 6}\) on the same set of axes. Note that the graph of a function and its inverse are reflections of each other about the line \(y = x\). Thus, plot this line on the same axes to check the validity of your graph. Use a variety of points for each function, including, if possible, the point (x, f(x)) from the original function and the point (f^{-1}(x), x) from the inverse function.

Step 4: Verify Correctness

Finally, it is always a good idea to verify your answer. Cross check the relationship between the function and its inverse: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). If both equations hold true, the calculated inverse is correct.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions with a degree of three. The general form is expressed as: \( f(x) = ax^3 + bx^2 + cx + d \). Here, \(a, b, c\), and \(d\) are constants, with \(a eq 0\). Cubic functions have a unique S-shape curve when graphed.
They exhibit several features:

• A single point of inflection where the curve changes concavity.
• Up to three real roots, which are the x-intercepts of the graph.
• Possibility of local maximum and minimum points depending on the sign and value of coefficients.

In the function \(f(x) = x^3 - 6\), we see a simple cubic function where the leading coefficient \(a = 1\), resulting in an upward curve. The point of inflection in this function would be at the origin were it not shifted vertically. Cubic functions can model phenomena with non-linear growth, such as in physics or economics, making them crucial in various applications.

Graphing

Graphing functions gives us a visual representation of how they behave. For cubic functions like \(f(x) = x^3 - 6\), we start by plotting points for several values of \(x\) to ensure accuracy. This particular function is vertically shifted down by 6 units because of the \(-6\) constant, which impacts the location of the graph on the y-axis.
The graph of a cubic function generally has a smooth, continuous curve with an S-like shape. An important aspect when graphing is to consider symmetry and turning points:

• The point of inflection is a vital feature marking the section where the curve goes from concave down to concave up.
• The graph is odd, meaning it is symmetric about the origin in its unshifted form.

When also graphing the inverse \(f^{-1}(x) = \sqrt[3]{x + 6}\), include the line \(y = x\) as a reference. The inverse function will reflect across this line, showcasing the symmetry between a function and its inverse as you plot them together.

Function Transformation

Function transformation involves shifting, stretching, or flipping the graph of a function. In \(f(x) = x^3 - 6\), we observe a vertical translation. This means every point on the original \(x^3\) graph has been shifted downwards by 6 units.
Function transformations have several types, each altering the graph uniquely:

• Vertical shifts, where constants are added or subtracted from the function, altering its position on the y-axis.
• Horizontal shifts involve changes inside the function, affecting its x-values.
• Stretches and compressions modify the scale of the graph, either amplifying or reducing its appearance.
• Reflections change the direction of the graph across an axis.

Inverse functions demonstrate such transformations by swapping x and y, which is a reflection over the line \(y = x\). Understanding transformations empowers us to predict and sketch the function's new shape and position on a graph effectively.