Problem 80 - College Algebra

Question

Use \(f(x)=x^{3}+1\) and \(g(x)=\sqrt[3]{x-1}\). Find \((f \circ g)(x)\) and \((g \circ f)(x)\). Compare the two answers.

Step-by-Step Solution

Answer

Both \((f \circ g)(x)\) and \((g \circ f)(x)\) equal \(x\). They are identical.

Step 1: Understand Composite Functions

The notation \((f \circ g)(x)\) means \(f(g(x))\), which represents substituting \(g(x)\) into function \(f(x)\). Similarly, \((g \circ f)(x)\) stands for \(g(f(x))\), which involves substituting \(f(x)\) into function \(g(x)\). We'll tackle each separately.

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

First, substitute \(g(x) = \sqrt[3]{x-1}\) into \(f(x) = x^3 + 1\). Thus, \(f(g(x)) = f(\sqrt[3]{x-1})\). Now apply the function: \[f(g(x)) = (\sqrt[3]{x-1})^3 + 1.\] By the property of radicals and exponents, \((\sqrt[3]{x-1})^3 = x-1\). So,\[f(g(x)) = x - 1 + 1 = x.\]

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

Now substitute \(f(x) = x^3 + 1\) into \(g(x) = \sqrt[3]{x-1}\). Thus, \(g(f(x)) = g(x^3 + 1)\). Now apply the function:\[g(f(x)) = \sqrt[3]{(x^3 + 1) - 1}.\] Simplify inside the cube root:\[g(f(x)) = \sqrt[3]{x^3} = x.\]

Step 4: Compare \((f \circ g)(x)\) and \((g \circ f)(x)\)

We found that \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\). Both compositions result in the identity function \(x\). Therefore, they are equal, which is a noteworthy property in this context.

Key Concepts

Function Composition

Function composition is an essential concept that involves combining two functions into one. If you have two functions, say \( f(x) \) and \( g(x) \), you can create a new function by plugging one function into the other. This process is denoted as \( (f \circ g)(x) \) for substituting \( g(x) \) into \( f(x) \), and \( (g \circ f)(x) \) for plugging \( f(x) \) into \( g(x) \).
For example, if \( f(x) = x^3 + 1 \) and \( g(x) = \sqrt[3]{x-1} \), then to find \( (f \circ g)(x) \), you substitute \( g(x) \) into every \( x \) in \( f(x) \). Thus, \( f(g(x)) = \left(\sqrt[3]{x-1}\right)^3 + 1 \). Similarly, \( (g \circ f)(x) \) would involve substituting \( f(x) \) into \( g(x) \), so \( g(f(x)) = \sqrt[3]{(x^3 + 1) - 1} \).
This concept is vital in helping understand how functions interact with each other and the outcomes of their compositions.

Identity Function

The identity function is a fundamental concept in the world of mathematics. It is a function that always returns the input given to it. In mathematical terms, it is represented as \( f(x) = x \).

When composing functions, if the result of the composition is the identity function, it means that the two functions, when combined, simply output the original input. In the example of the given exercise, both \((f \circ g)(x)\) and \((g \circ f)(x)\) result in \( x \). This indicates that their composition is the identity function.

• This property is useful in examining the reversibility of functions - whether applying one function and then another returns you to your starting point.
• Identity functions play a crucial role in understanding inverse functions and transformations within mathematical structures.

Grasping this simple yet powerful concept can greatly enhance one's comprehension of function operation and manipulation.

Cubic Functions

Cubic functions are polynomials with the highest degree of three, often in the format \( f(x) = ax^3 + bx^2 + cx + d \). Here, our function \( f(x) = x^3 + 1 \) is a simple cubic function without the \( x^2 \) and \( x \) terms.

• Cubic functions can have up to three real roots and display a unique S-shaped curve when graphed.
• They are used to model various types of natural and man-made phenomena, such as projectile motion and economic models.

In the context of composing functions, understanding the properties of cubic functions is key to predicting how they will interact when combined with other types of functions.
In the exercise, the interaction between the cubic function \( f(x) \) and the radical function \( g(x) \) reveals how compositions simplify back to the identity function, which underscores the beauty of function behavior.

Radicals and Exponents

Radicals and exponents are versatile and interconnected mathematical tools. Radicals are the opposite operation to exponents. For example, the cube root \( \sqrt[3]{x} \) is the inverse operation to raising something to the power of three, \( x^3 \).

In the context of function composition, understanding how radicals and exponents interact is crucial. Consider the function \( g(x) = \sqrt[3]{x-1} \). This radical function reverses the effect of the cubic function \( f(x) = x^3 + 1 \).

• When composed as \( f(g(x)) = \left(\sqrt[3]{x-1}\right)^3 + 1 \), the cube and cube root cancel each other, simplifying to \( x \).
• Alternatively, \( g(f(x)) = \sqrt[3]{(x^3 + 1) - 1} \) also resolves to \( x \) as the cube root and the cubic function undo each other, returning to the original value \( x^3 \).

This demonstrates how powerful and simplifying these operations can be when combined appropriately, particularly in achieving the identity function within compositions.