Problem 64 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Sketch the graph of \(f .(b)\) Use the graph of \(f\) to sketch the graph of \(f^{-1} .(c)\) Find \(f^{-1} .\) $$ f(x)=x^{3}-1 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step 1: Understand the Function

The given function is \( f(x) = x^3 - 1 \). This function is a cubic polynomial, which means its graph will be a curve that resembles an S-shape, with one point of inflection.

Step 2: Sketch the Graph of f(x)

To sketch \( f(x) = x^3 - 1 \), recognize that it is the graph of \( x^3 \) translated 1 unit downwards. Key points are \( f(0) = -1 \), \( f(1) = 0 \), and \( f(-1) = -2 \). The graph crosses the y-axis at -1 and has an inflection point at (0, -1).

Step 3: Reflect the Graph Across the Line y=x

To sketch \( f^{-1}(x) \), reflect the graph of \( f(x) = x^3 - 1 \) across the line \( y = x \). This reflection will produce the graph of the inverse function.

Step 4: Find the Inverse Function

To find \( f^{-1} \), start with \( y = x^3 - 1 \), solve for \( x \) in terms of \( y \): add 1 to both sides to get \( y + 1 = x^3 \), then take the cube root of both sides to find \( x = \sqrt[3]{y + 1} \). Thus, swap \( x \) and \( y \) to get \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step 5: Sketch the Graph of f^{-1}(x)

Now, sketch the graph of \( f^{-1}(x) = \sqrt[3]{x + 1} \). This graph should mirror the shape of \( f(x) = x^3 - 1 \) but flipped over the line \( y = x \). Key points are reversed: for instance, if \( f(1) = 0 \), then \( f^{-1}(0) = 1 \).

Key Concepts

Understanding Cubic Functions

Cubic functions are polynomial functions of degree three, and they typically have the form \( f(x) = ax^3 + bx^2 + cx + d \). In the given problem, we deal with a cubic function \( f(x) = x^3 - 1 \). This expression signifies that the highest power of \( x \) is three, indicating a cubic nature. The graph of a cubic function often has a characteristic "S" shape, which includes one point of inflection.

This point of inflection occurs where the concavity of the graph changes. For the function \( x^3 - 1 \), this point is at (0, -1). These graphs might not always intersect the x-axis three times, but they often exhibit three roots, where they touch or cross the x-axis.

Cubic functions are important to understand because they model various real-world phenomena where the relationship between variables isn't linear.

• Their unique curvature is especially helpful in physics and engineering to describe motion and change.
• Understanding the basics of cubic functions aids further exploration into higher-degree polynomials and more complex mathematical concepts.

Exploring Graph Reflection

Graph reflection is a useful concept in mathematics that helps in understanding how graphs of inverse functions relate to each other. Reflection involves flipping a graph over a specific line, typically the line \( y = x \) when dealing with inverse functions.

When you reflect a graph across the line \( y = x \), you essentially swap the x and y coordinates of points on the graph.
This transformation allows us to visualize the inverse of a given function in terms of how it "mirrors" the original function. In the given problem, reflecting the graph of \( f(x) = x^3 - 1 \) helps us sketch its inverse, \( f^{-1}(x) = \sqrt[3]{x + 1} \).

• Reflection makes it easier to see symmetric relationships between a function and its inverse.
• Key points such as intersections and inflection points can be reflected directly, helping in correctly sketching the inverse.
• This technique is especially valuable in exploring functions graphically when algebraic inversion is complex or impractical.

Understanding Polynomial Functions

Polynomial functions are a broad class of functions known for their structure and wide range of applications. They include any function that can be expressed in the form \( a_nx^n + a_{n-1}x^{n-1} + \, ...\, + a_1x + a_0 \), where each coefficient \( a \) is a constant, and \( n \) is a non-negative integer.

Cubic functions, like \( f(x) = x^3 - 1 \), are a specific type of polynomial where the highest power is three, but polynomial functions span a vast array of complex curves and patterns.

Polynomial functions can model simple to very complex behaviors in mathematical terms.

• Roots and zeroes provide key insights into where the graph intersects the x-axis – crucial for solving equations.
• The degree of the polynomial indicates the maximum number of roots and can help predict graph behavior throughout its domain.
• Understanding these functions aids in grasping more advanced calculus concepts, like derivatives and integrals.

Embracing the logical structure of polynomials enhances problem-solving skills and expands analytical thinking required for higher math.