Problem 45 - Precalculus Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x+1}\). Graphing both \(f(x)\) and \(f^{-1}(x)\), we find that they're mirror images of each other about the line \(y=x\). Finally, the domain and range for both \(f\) and \(f^{-1}\) are the set of all real numbers, denoted as \(-\infty < x, y < \infty \).

Step 1: Find the Inverse Function

Initially, replace \(f(x)\) with \(y\). This gives us the equation \(y=x^{3}-1\). Next, swap \(x\) and \(y\), this gives us \(x=y^{3}-1\). Now solve for \(y\) which will give us the inverse function. Add \(1\) to both sides and take cube root, resulting in \(y=\sqrt[3]{x+1}\). Therefore the inverse function \(f^{-1}(x)=\sqrt[3]{x+1}\)

Step 2: Graph the function and its Inverse

Plot the function \(y=x^3-1\), which is a cubic function shifted down by \(1\). Next, plot \(f^{-1}(x) = \sqrt[3]{x+1}\) which will be a reflection of the graph of \(f(x)\) about the line \(y=x\). The result should be two graphs that are mirror images of each other across the line \(y = x\).

Step 3: Determine the Domain and Range

For the function \(f(x)=x^3-1\), it's a polynomial. Thus, the domain (set of possible input \(x\)-values) is all real numbers \(-\infty < x < \infty\). The range for \(f(x)\) (possible output values) is also all real numbers since it's a cubic function which continues indefinitely in both positive and negative \(y\) directions. For \(f^{-1}(x) = \sqrt[3]{x+1}\), the situation is the same because cubic root can also accept any real number. Thus, the domain and range of both functions is \(-\infty < x, y < \infty\).

Key Concepts

Graphing Functions

Understanding how to visualize mathematical relationships is essential for students, and graphing functions is a foundational skill in this regard. When graphing functions such as cubic functions, it is crucial to recognize the shape and behavior of the graph.

For a basic cubic function, such as f(x) = x^3, the graph starts in the bottom left quadrant, swoops through the origin, and proceeds into the top right quadrant, creating an S-like curve. Any transformations such as shifting, reflecting, or stretching can adjust this fundamental shape. When graphing, start by plotting critical points like intercepts and then use the nature of the function to determine additional points. In the case of the given exercise, graphing f(x) = x^3 - 1 involves a vertical shift. Practice by plotting several values of x to get a visual curve of the function.

Remember to draw the graph smoothly with continuous lines as cubic functions do not have breaks or corners. Also, include the reflection of the function about the line y = x to represent the inverse function, which helps in visualizing the concept of inverse relations.

Domain and Range

The domain and range of a function form the cornerstone of function analysis and should be understood deeply. The domain represents all the possible inputs, or x-values, that a function can accept without causing undefined outputs. For cubic functions, which are polynomial functions with odd degrees, the domain is always all real numbers, expressed as \( -\infty, \infty \). The range is the set of all possible outputs, or y-values, that the function can produce. For basic cubic functions, the range is also all real numbers, since as x increases or decreases, the output can span from negative to positive infinity.

Reflecting this understanding in notation, you will often describe the domain and range using interval notation. For the exercise's cubic function, we state the domain and range as \( -\infty, \infty \), signifying that any real number is a valid input and output. The same applies to its inverse function since reflection over the line y = x does not alter the 'completeness' of the function. This illustrates an important property: for functions that are inverses of each other, the domain of one is the range of the other and vice versa.

Cubic Functions

Cubic functions are a class of polynomial functions where the highest degree of any term is three, typically written in the form f(x) = ax^3 + bx^2 + cx + d. The nature of a cubic function's graph is characterized by its 'S' shaped curve, which can experience translations, reflections, and stretching based on the coefficients and constant terms.

For example, the exercise provided focused on the function f(x) = x^3 - 1. This particular cubic function is a simple case where a=1, b=0, c=0, and d=-1. The -1 signifies a downward shift of the graph. Cubic functions have notable features: they have no bounded extremas since they extend to infinity in both directions, and their end behavior contrasts at each end of the graph. One tail will approach negative infinity while the other approaches positive infinity, as x increases and decreases respectively.

When analyzing cubic functions, pay special attention to their intercepts, rates of increase, and decrease, and how they change before and after their points of inflection, where the curvature changes direction. These characteristics will help you sketch accurate graphs and gain a deeper understanding of cubic functions' properties.