Problem 27 - Algebra Form and Function

Question

Find (a) \(\quad f(g(x))\) (b) \(g(f(x))\) $$ f(x)=x^{3} \text { and } g(x)=5+2 x $$

Step-by-Step Solution

Answer

Based on the step-by-step solution, find the composite functions \(f(g(x))\) and \(g(f(x))\). f(g(x)) = 8x^3 + 60x^2 + 150x + 125 g(f(x)) = 2x^3 + 5

Step 1: Understanding Composite Functions

A composite function is a function created by applying one function to the outputs of another. Given two functions \(f(x)\) and \(g(x)\), the composite function \(f(g(x))\) is denoted as \(f \circ g\), and can be found by replacing every \(x\) in the function \(f(x)\) with the function \(g(x)\). Similarly, the composite function \(g(f(x))\) is denoted as \(g \circ f\), and can be found by replacing every \(x\) in the function \(g(x)\) with the function \(f(x)\).

Step 2: Finding the Composite Function \(f(g(x))\)

To find the composite function \(f(g(x))\), we will substitute the function \(g(x) = 5+2x\) into \(f(x)=x^3\). This means that wherever there is an 'x' in the \(f(x)\) function, we will replace it with the entire expression of \(g(x)\). $$ f(g(x)) = f(5 + 2x) = (5 + 2x)^3 $$ Now, let's expand the expression \((5 + 2x)^3\).

Step 3: Expanding the Expression \((5 + 2x)^3\)

To expand the expression \((5 + 2x)^3\), we will apply the binomial theorem. This theorem tells us that the expansion of a binomial (a + b) raised to a power n is given by: $$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$ Using this formula, we can expand \((5+2x)^3\): $$ (5+2x)^3 = \binom{3}{0} 5^{3} (2x)^0 + \binom{3}{1} 5^{2} (2x)^1 + \binom{3}{2} 5^{1} (2x)^2 + \binom{3}{3} 5^{0} (2x)^3 $$ Now, let's compute the coefficients and simplify the expression.

Step 4: Computing the Coefficients and Simplifying the Expression

Using the binomial coefficients, we can simplify the expression: $$ (5 + 2x)^3 = 1 \cdot 125 \cdot 1 + 3 \cdot 25 \cdot 2x + 3 \cdot 5 \cdot 4x^2 + 1 \cdot 1 \cdot 8x^3 $$ Finally, let's add the terms together: $$ (5 + 2x)^3 = 125 + 150x + 60x^2 + 8x^3 $$ Now we have found the composite function \(f(g(x))\): $$ f(g(x)) = 8x^3 + 60x^2 + 150x + 125 $$

Step 5: Finding the Composite Function \(g(f(x))\)

Similar to step 2, to find the composite function \(g(f(x))\), we will substitute the function \(f(x) = x^3\) into \(g(x)=5+2x\). This means that wherever there is an 'x' in the \(g(x)\) function, we will replace it with the entire expression of \(f(x)\). $$ g(f(x)) = g(x^3) = 5 + 2(x^3) $$ Now, let's simplify the expression.

Step 6: Simplifying the Expression \(5 + 2(x^3)\)

The expression \(5 + 2(x^3)\) can be rewritten as: $$ g(f(x)) = 5 + 2x^3 $$ Now, we have found the composite function \(g(f(x))\): $$ g(f(x)) = 2x^3 + 5 $$

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra for expanding expressions that are raised to a power. It provides a formula that helps us break down the problem of expanding expressions like \((a + b)^n\) into a sum of terms. Each term in the expansion has a coefficient called the binomial coefficient, which can be calculated using combinations.

• For any two terms \(a\) and \(b\), and a positive integer \(n\), the binomial expansion is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
• The binomial coefficient \(\binom{n}{k}\) indicates the number of ways to choose \(k\) items from \(n\) items and is computed as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
• This method is particularly useful when the expressions involve powers higher than two, like our exercise where \((5 + 2x)^3\) was expanded.

By using the binomial theorem, we efficiently expanded \((5 + 2x)^3\) into individual terms, revealing more understandable expressions suitable for further algebraic manipulations.

Function Composition

Function composition is a process where one function is applied to the result of another function. It allows creating new functions from known ones by feeding the output of one function into another.

• When two functions \(f(x)\) and \(g(x)\) are given, the composite function \(f(g(x))\) involves substituting the entire expression of \(g(x)\) into \(f(x)\).
• This is denoted as \(f \circ g\), meaning you start with \(g(x)\) and then use \(f(x)\) on it.

In the original exercise, we explored two composite functions:

• To find \(f(g(x))\), we replaced the \(x\) in \(f(x) = x^3\) with the entire expression \(g(x) = 5 + 2x\), resulting in \(f(g(x)) = (5 + 2x)^3\).
• Similarly, for \(g(f(x))\), we substituted \(x^3\) from \(f(x) = x^3\) into \(g(x) = 5 + 2x\), yielding \(g(f(x)) = 5 + 2x^3\).

Composing functions this way not only allows us to combine different operations but also streamlines complex calculations and seeks new patterns or behaviors of inputs.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operators. They form the foundation of algebra and provide ways to represent real-world problems.

• An algebraic expression might be something simple like \(2x + 3\) or more complex like \(5 + 2x^3\), as seen in the exercise.
• These expressions can be manipulated following systematic rules, such as expanding, factoring, and simplifying.

In the exercise, algebraic expressions were vital in solving the problem of finding composite functions:

• The expression \(f(g(x)) = (5 + 2x)^3\) was expanded using the binomial theorem.
• For \(g(f(x)) = 5 + 2x^3\), no further simplification was needed, but the problem illustrated how algebra operates on expressions with different terms.

Understanding how to work with algebraic expressions effectively opens up countless possibilities for solving equations, analyzing functions, and modeling scenarios.