Problem 72 - A Graphical Approach to College Algebra

Question

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for the pairs. $$f(x)=\sqrt[3]{x+1}, g(x)=x^{3}-1$$

Step-by-Step Solution

Answer

The pairs \(f(x)=\sqrt[3]{x+1}\) and \(g(x)=x^{3}-1\) satisfy \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\).

Step 1: Understand Function Composition

Function composition involves applying one function to the result of another function. Here we need to verify that for the pairs \(f\) and \(g\), the composition \((f \circ g)(x)\) and \((g \circ f)(x)\) both result in \(x\).

Step 2: Compute \((f \circ g)(x)\)

The function \(f\) is \(f(x)=\sqrt[3]{x+1}\) and \(g\) is \(g(x)=x^{3}-1\). Find \((f \circ g)(x)\) by replacing \(x\) in \(f(x)\) with \(g(x)\):\[(f \circ g)(x) = f(g(x)) = f(x^{3}-1) = \sqrt[3]{(x^{3}-1) + 1} = \sqrt[3]{x^{3}}.\]\(\sqrt[3]{x^{3}} = x.\) Thus, \((f \circ g)(x) = x\).

Step 3: Compute \((g \circ f)(x)\)

Now calculate \((g \circ f)(x)\), where \(g(x)=x^{3}-1\) and \(f(x)=\sqrt[3]{x+1}\). Replace \(x\) in \(g(x)\) with \(f(x)\):\[(g \circ f)(x) = g(f(x)) = g(\sqrt[3]{x+1}) = (\sqrt[3]{x+1})^3 - 1.\]This simplifies to: \((\sqrt[3]{x+1})^3 = x+1.\) Hence, \[(g \circ f)(x) = x+1 - 1 = x.\]So, \((g \circ f)(x) = x\).

Step 4: Conclusion

Both compositions \((f \circ g)(x)\) and \((g \circ f)(x)\) resulted in \(x\). Therefore, the functions \(f\) and \(g\) indeed satisfy the requirement of \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\).

Key Concepts

Inverse Functions

Inverse functions are intriguing components of mathematics because they allow us to reverse a process or calculation performed by a given function. An inverse function essentially "undoes" what the original function does. When you have a function \(f(x)\) and its inverse \(f^{-1}(x)\), applying one and then the other will yield the starting value, \(x\), such that \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\). This process is particularly useful in many areas of algebra where checking for the existence of an inverse can provide insights into the function's properties.

To confirm that two functions are inverses of each other, you compute both their compositions, \((f \circ g)(x)\) and \((g \circ f)(x)\), and check if both results simplify to \(x\). This is how we verify the pairs in our example, showing the two are indeed inverses. Having an intuition about inverse functions strengthens the foundation of understanding algebraic manipulation.

Algebraic Functions

Algebraic functions are those constructed using basic algebraic operations such as addition, subtraction, multiplication, division, and exponentiation with integer powers. Understanding how algebraic functions work is crucial since they are everywhere in mathematics, and simplifying these functions is a common task.

Functions like polynomials and rational functions fall under this category. In the given example, \(g(x) = x^3 - 1\) is an algebraic function, since it involves polynomial operations which are foundational to algebra. Recognizing and working with algebraic operations allows students to solve complex problems by breaking them down into simpler, manageable parts.

• Polynomials, like cubic expressions, reveal much about the nature of functions, including their inverses.
• Simplifying algebraic expressions involves understanding distributive, associative, and commutative properties.

In-depth knowledge of these functions aids students in understanding broader mathematical concepts that build upon basic algebraic principles.

Cubic Root Function

The cubic root function, represented as \(f(x) = \sqrt[3]{x} \), is an invaluable tool for solving equations where a variable is raised to the power of three. Unlike square root functions, cubic roots apply to all real numbers, including negative numbers since any real number has a real cubic root. This makes cubic roots particularly versatile in algebra.

The specific function \(f(x) = \sqrt[3]{x+1} \) operates by taking any input, increasing it by one, and then applying the cubic root to the result. Because cube roots can handle negative and positive inputs gracefully, they exhibit different characteristics than other rooting functions.

• Cubic roots provide symmetry in calculations since \( \sqrt[3]{a^3} = a \) holds for all real \(a\).
• They are crucial in simplifying expressions involving cube powers.

Understanding the behavior of cubic root functions allows students to effectively manipulate and solve equations involving third-degree expressions, which often appear in real-world applications.