Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{3}+1$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=x^{3}+1\) is \(f^{-1}(x) = \sqrt[3]{x-1}\). When both functions are graphed, they are symmetrical about the line \(y=x\).

Step 1: Write the function

Let's begin by writing the given function \(f(x)=x^{3}+1\) where \(y=f(x)\). So, we can rewrite the function as \(y=x^{3}+1\).

Step 2: Find the inverse function

To find the inverse, swap \(x\) and \(y\) to get \(x=y^{3}+1\). Then solve for \(y\) to find the inverse function. Subtract 1 from both sides so that expression becomes \(x-1=y^{3}\). Then take cubic root from both sides to solve for \(y\), so \(f^{-1}(x) = y = \sqrt[3]{x-1}\). Therefore, the inverse function of \(f(x)\) is \(f^{-1}(x) = \sqrt[3]{x-1}\)

Step 3: Graph and Analyze the functions

Next, use a graphing utility to graph both \(f(x)\) and \(f^{-1}(x)\) in the same viewing window. The graph of the original function \(f(x)\) is a cubic function which increases as \(x\) increases. The graph of \(f^{-1}(x)\) looks like a flipped version of the graph of \(f(x)\) over the line \(y=x\). The two graphs are symmetrical about the line \(y=x\) as is typical for a function and its inverse.

Key Concepts

Graphical Representation

Graphical representation of functions is a helpful way to visualize the relationship between a function and its inverse. By plotting both a function and its inverse on the same graph, we can clearly see how they interact. For a function like \(f(x) = x^3 + 1\), its graphical representation is a cubic curve which continuously increases or decreases based on the value of \(x\).
Its inverse, \(f^{-1}(x) = \sqrt[3]{x-1}\), would typically appear as a flipped or mirrored version of the original function.
A graphing utility can conveniently plot these curves. Observing them together in the same window allows us to determine their patterns quickly, helping us see intrinsic properties such as symmetry. This visual approach illuminates the connection between a function and its inverse much more vividly than algebraic manipulation alone.
Some key points to note in graphical representations are:

• The intersection points, which usually occur where the function equals its inverse.
• The overall shape and direction of growth or decline.

Cubic Functions

Cubic functions are polynomial functions of degree three, typically of the form \(f(x) = ax^3 + bx^2 + cx + d\). In the problem above, the function \(f(x) = x^3 + 1\) is a simple cubic function. Such functions have non-linear graphs that can either rise or fall dramatically, and they tend to have a characteristic 'S' or inverted 'S' shape depending on the coefficients.An important aspect of cubic functions is that they have an inflection point where the graph changes concavity. This can often result in cubic functions intersecting with their inverse along the line \(y=x\) in distinct patterns.
Understanding cubic functions involves:

• Recognizing the effect of each coefficient, especially the leading coefficient, on the graph's shape.
• Knowing how to find key features like intercepts and points of inflection.

For students learning algebra, cubic functions provide an exciting way to deepen their understanding of how polynomial functions behave and how to manipulate them both graphically and algebraically.

Symmetry in Graphs

Symmetry in graphs is a fundamental concept in understanding the relationship between functions and their inverses. When we graph both a function and its inverse, such as the ones in the exercise, we frequently find symmetry about the line \(y = x\). This line acts as a mirror, illustrating how one graph reflects into the other.This symmetry is a defining feature of inverse functions. It stems from the fact that swapping the roles of \(x\) and \(y\) coordinates flips the graph over this line. For instance, for \(f(x) = x^3 + 1\) and its inverse \(f^{-1}(x) = \sqrt[3]{x-1}\), their plots demonstrate this symmetry perfectly.Recognizing symmetry includes:

• Identifying the line of symmetry, which is \(y = x\) for inverse functions.
• Understanding how reflecting across the line \(y = x\) changes each \((x, y)\) point to its inverse \((y, x)\).

Exploring symmetry in graphs enhances the comprehension of the particular elegance in mathematical functions, showcasing the precision and balance inherent in algebraic relationships.