Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(g(x)=x^{2}+1\), find formulas for \(g^{3}(x)\) and \((g \circ g \circ g)(x)\).

Step-by-Step Solution

Answer

\(g^3(x) = x^6 + 3x^4 + 3x^2 + 1\); \(g(g(g(x))) = x^8 + 4x^6 + 8x^4 + 8x^2 + 5\).

Step 1: Understand the Notation

The expression \(g^{3}(x)\) means we want to compute \(g(x)\) to the power of 3, i.e., \([g(x)]^3\). The notation \((g \circ g \circ g)(x)\) represents the composition of the function \(g\) three times, which means applying \(g\) to \(x\), then applying \(g\) again to that result, and once more applying \(g\) to the new result.

Step 2: Compute \(g(x)^3\)

The function given is \(g(x) = x^2 + 1\). Therefore, \(g^3(x)\) can be computed as follows:\[g^3(x) = (x^2 + 1)^3\]You need to expand \((x^2 + 1)^3\) using the binomial theorem or manually by multiplying \(x^2 + 1\) three times.

Step 3: Expand \((x^2 + 1)^3\)

To expand \((x^2 + 1)^3\), we use the formula:\[(x^2 + 1)^3 = (x^2 + 1)(x^2 + 1)(x^2 + 1)\]First, find \((x^2 + 1)(x^2 + 1) = x^4 + 2x^2 + 1\), then multiply the result by \(x^2 + 1\):\[(x^4 + 2x^2 + 1)(x^2 + 1) = x^6 + 2x^4 + x^2 + x^4 + 2x^2 + 1 = x^6 + 3x^4 + 3x^2 + 1\]So, \(g^3(x) = x^6 + 3x^4 + 3x^2 + 1\).

Step 4: Find \(g(g(g(x)))\)

First, calculate \(g(g(x))\):\[g(g(x)) = g(x^2 + 1) = (x^2 + 1)^2 + 1 = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2\]Next, compute \(g(g(g(x)))\):\[g(g(g(x))) = g(x^4 + 2x^2 + 2) = (x^4 + 2x^2 + 2)^2 + 1\]You would need to expand \((x^4 + 2x^2 + 2)^2\) using similar methods to find this value, which results in:\[x^8 + 4x^6 + 8x^4 + 8x^2 + 4 + 1 = x^8 + 4x^6 + 8x^4 + 8x^2 + 5\]So, \(g(g(g(x))) = x^8 + 4x^6 + 8x^4 + 8x^2 + 5\).

Step 5: Review Results

The formulas derived from the function \(g(x) = x^2 + 1\) are:- \(g^3(x) = x^6 + 3x^4 + 3x^2 + 1\)- \(g(g(g(x))) = x^8 + 4x^6 + 8x^4 + 8x^2 + 5\)

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows you to expand expressions that have been raised to a power. It is particularly useful when dealing with polynomial expressions, such as \((x^n + a)^m\). In the context of the exercise, we used the binomial theorem to expand \((x^2 + 1)^3\).
It works by using specific coefficients, known as binomial coefficients, which are derived from Pascal's triangle. This theorem states that:

• The expression \((a + b)^n\) can be expanded as a sum of terms.
• Each term in the expansion follows the pattern of binomial coefficients \(\binom{n}{k}\).
• Which represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection.

In our problem, for example, the binomial expansion of \((x^2 + 1)^3\) was demonstrated step by step, resulting in the polynomial \(x^6 + 3x^4 + 3x^2 + 1\). This method helps simplify and solve complex algebraic problems efficiently.

Function Notation

Function notation is a way of representing functions in mathematics to make them easier to understand and use. It is commonly denoted by letters such as \(f(x)\), \(g(x)\), or \(h(x)\), where the letter represents the function and \(x\) is the variable. For example, in the problem at hand, \(g(x) = x^2 + 1\) indicates that if you plug a value into \(x\), you'll receive the value of \(x\) squared, plus 1.
This type of notation is crucial as it keeps mathematical expressions organized and makes it clear how different operations like evaluation, composition, and inversion should be carried out.
In our exercise, understanding the notation \(g^3(x)\) and \((g \circ g \circ g)(x)\) was an imperative step to find the formulas effectively. \(g^3(x)\) indicates a polynomial expansion of \(g(x)\), mentioned specifically as raising to the third power. Meanwhile, \((g \circ g \circ g)(x)\) represents a function composition, where the function is applied three successive times.

Polynomial Expansion

Polynomial expansion is the process of expanding expressions such as \((x + y)^n\) into a sum involving terms of the form \(ax^by^c\). When you expand, you basically distribute all terms in the expression to achieve a broader expression that combines like terms.
For example, let's look at expanding \((x^2 + 1)^3\):

• First, find \((x^2 + 1)(x^2 + 1) = x^4 + 2x^2 + 1\).
• Then, multiply this result by \((x^2 + 1)\) again to find the full expansion, resulting in \(x^6 + 3x^4 + 3x^2 + 1\).
  

In the problem, polynomial expansion isn't just applied to \((x^2 + 1)^3\), but it's crucial when finding \(g(g(g(x)))\). Repeatedly using polynomial expansion helps achieve the final form \(x^8 + 4x^6 + 8x^4 + 8x^2 + 5\). Understanding polynomial expansion is key when manipulating and simplifying expressions in algebra.