Problem 43 - 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 \(f^{-1} .\) \(f(x)=\sqrt[3]{x-1}\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x)=x^3+1\). Both the function and its inverse are graphically symmetric about the line \(y = x\), and their domain and range includes all real numbers.

Step 1: Find the inverse function

Recall that to find the inverse of a function, one can swap \(x\) and \(y\) and then solve for \(y\). So the first step is to exchange \(x\) and \(y\) in the equation \(y = \sqrt[3]{x-1}\) which gives \(x = \sqrt[3]{y-1}\). Then, cube both sides of the equation \(x = \sqrt[3]{y-1}\) to eliminate the cube root on the right side. After cubing both sides, the equation will be \(x^3 = y-1\). Then, finally solve for \(y\), the other side of the equation will be \(y = x^3 + 1\). So the inverse function \(f^{-1}(x)=x^3+1\).

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

Graphing functions require a basic understanding of how function graphs look. For the function \(f(x) = \sqrt[3]{x-1}\), the x-intercept is at \(x = 1\). For the inverse function \(f^{-1}(x) = x^3 + 1\), there is a y-intercept at \(y = 1\). The graph of a cubic function is symmetric with respect to the origin. When the two graphs are superimposed, they are symmetric about the line \(y = x\).

Step 3: Describe the relationship between the graphs

Looking at the graphs of both functions, one can observe they are symmetric about the line \(y = x\). This is a characteristic feature of a function and its inverse.

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

Recall that the domain of a function is the set of all possible x-values and the range is the set of all possible y-values. For the function \(f(x) = \sqrt[3]{x-1}\), the domain is all real numbers and the range is also all real numbers. For the inverse function \(f^{-1}(x) = x^3 + 1\), the domain and range are also all real numbers.

Key Concepts

Function Graphing

Graphing functions helps visualize the relationship between variables. When graphing the function \(f(x) = \sqrt[3]{x-1}\), one key feature to note is the x-intercept at \(x = 1\). This is where the graph crosses the x-axis. For the inverse, \(f^{-1}(x) = x^3 + 1\), focus on the y-intercept at \(y = 1\), where the graph crosses the y-axis.
These intercepts provide anchor points for sketching the curves.
When plotting, observe that both graphs extend infinitely without bounds. This behavior signifies that there are no restrictions on the values x and y can take.
As you map these functions onto coordinate axes, look for smooth transitions indicating the continuous nature of the functions.

• Graph intersects: x-intercept at \(x = 1\), y-intercept at \(y = 1\) for the inverse.
• Smooth curve, infinite extension.

Domain and Range

Understanding the domain and range clarifies the behavior of real functions. For \(f(x) = \sqrt[3]{x-1}\), the domain is all real numbers \((-\infty, \infty)\) because a cube root is defined for any real number.
The range is likewise all real numbers, as the output of the function can span any value.
Similarly, \(f^{-1}(x) = x^3 + 1\) also has a domain and range of all real numbers. This is due to the nature of the cubic function, allowing for an unrestricted input and output.
Recognizing unlimited domains and ranges is crucial for understanding how these functions behave in real-world applications.

• Domain of \(f(x)\): All real numbers \((-\infty, \infty)\).
• Range of \(f(x)\): All real numbers \((-\infty, \infty)\).
• Domain of \(f^{-1}(x)\): All real numbers \((-\infty, \infty)\).
• Range of \(f^{-1}(x)\): All real numbers \((-\infty, \infty)\).

Symmetric Properties

The symmetry of graphs is an intrinsic aspect of inverse functions. For any function and its inverse, such as \(f(x)\) and \(f^{-1}(x)\), their graphs will always mirror each other about the line \(y = x\).
This symmetry implies that all points \((a, b)\) on the function correspond to points \((b, a)\) on its inverse.
The line \(y = x\) acts as a diagonal axis of symmetry, highlighting the inherent relationship between the function and its inverse.
With cubic functions in particular, this symmetry is noticeable and makes it easier to verify whether a calculated inverse is correct.

• Mirroring around \(y = x\).
• Points \((a, b)\) on \(f\) correspond to \((b, a)\) on \(f^{-1}\).

Cubic Functions

Cubic functions like \(y = x^3\) have distinctive properties that differentiate them from linear or quadratic functions. The general form creates an S-shaped curve that is symmetric about the origin.
For \(f(x) = \sqrt[3]{x-1}\), this transformation shifts the graph, maintaining its cubic nature but altering its position.
Cubic functions grow more rapidly as the variable \(x\) increases or decreases from zero, displaying a steep, non-parabolic curve.
Understanding these properties aids in predicting the movement and behavior of cubic graphs across coordinate axes.

• Cubic function exhibits symmetry about origin.
• Swift growth as \(x\) increases/decreases.
• Distinct S-shaped curve.