Problem 80 - College Algebra

Question

For the following exercises, 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 compositions, \\( (f \circ g)(x) \\) and \\( (g \circ f)(x) \\), equal \\(x\\).

Step 1: Understanding the Composition of Functions

To solve the problem, we need to find the composition of functions. This means we have to evaluate two functions: \( (f \circ g)(x) \) and \( (g \circ f)(x) \). Here, \( (f \circ g)(x) \) means applying \(g(x)\) first and then applying \(f(x)\), while \( (g \circ f)(x)\) means applying \(f(x)\) first and then applying \(g(x)\).

Step 2: Finding (f ∘ g)(x)

First, we substitute \(g(x) = \sqrt[3]{x-1}\) into \(f(x)\), which gives us: \[(f \circ g)(x) = f(\sqrt[3]{x-1})\]. Substitute into \(f(x) = x^3 + 1\): \[f(\sqrt[3]{x-1}) = (\sqrt[3]{x-1})^3 + 1\]. The cube and cube root cancel out, so: \[f(\sqrt[3]{x-1}) = x - 1 + 1 = x\].

Step 3: Finding (g ∘ f)(x)

Next, substitute \(f(x) = x^3 + 1\) into \(g(x)\), which gives us: \[(g \circ f)(x) = g(x^3 + 1)\]. Substitute into \(g(x) = \sqrt[3]{x-1}\): \[g(x^3 + 1) = \sqrt[3]{(x^3 + 1) - 1} = \sqrt[3]{x^3}\]. The cube root and cube cancel each other, leaving us with: \[g(x^3 + 1) = x\].

Step 4: Comparing the Two Results

Both results \( (f \circ g)(x) = x \) and \( (g \circ f)(x) = x \) resulted in the same expression, which means that the two compositions of the functions are equal. In this case, \( f \circ g \) and \( g \circ f \) are equal, both simplifying to the identity function \(x\).

Key Concepts

Identity function

An identity function is a simple but fundamental concept in mathematics. It is a function that always returns the same value that was used as its input. For example, if you input any number into the identity function, you'll get that number back as the output. In mathematical terms, this is expressed as \( f(x) = x \).

Identifying when a function composition results in an identity function is essential. It shows that two functions can "undo" each other in a specific sequence. In the exercise you've seen, both combinations \((f \circ g)(x)\) and \((g \circ f)(x)\) simplify to \( x \) after applying function operations. This result indicates that they act like identity functions in their operation because they return the input value without alteration.

Whenever you encounter composed functions leading to an identity, it means there is a nice symmetry between the two functions. They perfectly balance each other out, giving back values that maintain their original form.

Function operations

Function operations involve various ways to combine and manipulate functions. These operations include addition, subtraction, multiplication, division, and composition of functions. In the context of the exercise, composition is key.

Composition of functions entails substituting one function into another. This is a two-step process. First, take the output of the first function you apply and use it as the input for the second function. Think of it as a conveyor belt where the output of one ends up as the material for the next operation.

• To compute \((f \circ g)(x)\), start with the input \(x\), apply \(g\), and then apply \(f\) on the result.
• For \((g \circ f)(x)\), begin with \(x\), apply \(f\), and then \(g\) on that result.

In the example provided, applying these operations directly led to simplification to the identity function, \(x\), indicating perfect interchangeability under composition for these specific functions.

Cubic functions

Cubic functions are polynomials of degree three and take the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). In this exercise, we deal with the cubic function \( f(x) = x^3 + 1 \).

Cubic functions can produce interesting results when composed with other functions because of their non-linear nature. They are versatile and can create a variety of curves and distinct shapes on a graph. These functions often appear in the real world as they can model a range of phenomena that show a three-dimensional structure or growth.

When a cubic function is involved in function operations such as composition, it's essential to understand how its properties combine with the properties of the other function. Here, by combining \( f(x) \) with \( g(x) = \sqrt[3]{x-1} \), the exercise shows how their composition simplifies beautifully to an identity function, demonstrating the unique interactions that occur with cubic functions. When composing, especially with cube roots, the processes often lead to revealing harmonic outcomes, like reaching back to the linear identity \(x\).