Problem 31 - Contemporary Precalculus

Question

Verify that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\) for every \(x\) $$f(x)=\sqrt[3]{x}+2, \quad g(x)=(x-2)^{3}$$

Step-by-Step Solution

Answer

Short Answer: Yes, the functions f(x) and g(x) are inverse functions of each other because both (f ∘ g)(x) and (g ∘ f)(x) equal x for every x.

Step 1: Find (f ∘ g)(x)

To find the composition (f ∘ g)(x), we need to substitute g(x) into the function f(x): (f ∘ g)(x) = f(g(x)). $$f(g(x)) = f((x-2)^{3})$$ Now, plug in the expression for f(x): $$f(g(x)) = \sqrt[3]{(x-2)^{3}}+2$$

Step 2: Simplify (f ∘ g)(x)

We now need to simplify the expression from Step 1 by using the properties of radicals. $$\sqrt[3]{(x-2)^{3}}+2 = x-2+2$$ The cube root and the cube cancel each other out, and the equation simplifies to: $$x$$

Step 3: Find (g ∘ f)(x)

To find the composition (g ∘ f)(x), we need to substitute f(x) into the function g(x): (g ∘ f)(x) = g(f(x)). $$g(f(x)) = g(\sqrt[3]{x}+2)$$ Now, plug in the expression for g(x): $$g(f(x)) = (\sqrt[3]{x}+2-2)^{3}$$

Step 4: Simplify (g ∘ f)(x)

We now need to simplify the expression from Step 3 by using the properties of radicals. $$(\sqrt[3]{x}+2-2)^{3} = (\sqrt[3]{x})^{3}$$ The terms +2 and -2 cancel out, leaving us with: $$x$$ Since both (f ∘ g)(x) and (g ∘ f)(x) equal x for every x, we can conclude that the given functions f(x) = √[3]{x} + 2 and g(x) = (x - 2)^3 are indeed inverse functions of each other.

Key Concepts

Composition of Functions

Composition of functions is a fundamental concept where two functions are combined to form a new function. It involves applying one function to the results of another function. If you have two functions, say \( f(x) \) and \( g(x) \), their composition is denoted as \((f \circ g)(x)\), meaning you first apply \( g(x) \) and then apply \( f(x) \) to the result.
For example, in our problem, when finding \((f \circ g)(x)\), we substitute \(g(x)\) into \(f(x)\). Conversely, for \((g \circ f)(x)\), we substitute \(f(x)\) into \(g(x)\).

• First, apply \(g(x)\): \(g(x) = (x-2)^3\)
• Then, apply \(f(x)\): \(f(x) = \sqrt[3]{x} + 2\)

This sequential operation shows how one output becomes the input for the next function, and through this process, we can check if these functions are inverses by evaluating \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\) for every \(x\).
When these conditions are met for all values in the domain, the functions are indeed inverses of each other.

Properties of Radicals

Radicals, such as square roots or cube roots, follow specific properties that help simplify expressions. A critical property is the fact that radicals and exponents can "undo" each other. For instance, the cube root of a number cubed equals the original number. This property is key in our problem.
Let's use this property: for \(x^3\) we have the cube root \(\sqrt[3]{x^3} = x\).
In our exercise, we simplify the expression \( \sqrt[3]{(x-2)^3} + 2\), where the cube root and cubing cancel each other out:

• The cube root \(\sqrt[3]{()}\) gets rid of the cube \(()^3\), leaving us with \(x-2\)
• Adding \(2\) results in the final simplification to \(x\)

Understanding these properties allows you to tackle more complex problems with confidence. Recognizing how radicals work will make algebraic manipulations smoother and often less daunting.

Function Simplification

Simplifying functions is essential in finding inverse functions and navigating complex equations. It involves breaking down expressions into their simplest form, which often means removing any unnecessary or redundant components. In the context of this exercise, simplification involves clearing radicals or certain algebraic terms.
Here, the simplification process let us take \((f \circ g)(x) = \sqrt[3]{(x - 2)^{3}} + 2\), which becomes \(x\) after simplification because:

• The cube \(()^3\) and cube root \(\sqrt[3]{()}\) cancel
• The terms \(+2\) and \(-2\) cancel each other

Resulting in the expression \(x\), keeping the integrity of both functions and allowing us to see if they are truly inverse functions.
Simplifying effectively is valuable as it lets you see results more clearly and can often lead to surprising simplifications, showing a deeper relationship between functions.

Algebraic Manipulation

Algebraic manipulation is like solving a puzzle where you move, combine, or cancel terms to find a solution. This technique is used heavily in solving equations and verifying properties of functions. In this problem, algebraic manipulation helps confirm the inverse relationship between \(f(x)\) and \(g(x)\).
Steps taken include:

• Combining like terms: such as \(+2\) and \(-2\), which cancel out
• Using the cube and cube roots to "cancel" and simplify the expression
• Rearranging parts of an equation to achieve a direct expression

These steps lead to validation that \((f \circ g)(x)\) and \((g \circ f)(x)\) both simplify to \(x\). Analyzing and rearranging algebraic expressions can illuminate key insights that aren't immediately visible.
Algebraic manipulation not only simplifies but often reveals deeper connections between mathematical entities, which is crucial in understanding inverses and broader mathematical contexts.