Problem 11 - Intermediate Algebra

Question

Complete each solution. Let \(f(x)=3 x-1\) and \(g(x)=2 x+3 .\) Find \(f \cdot g\). $$ \begin{aligned} (f \cdot g)(x) &=f(x) \cdot \square \\ &\left.=\square (2 x+3\right) \\ &=6 x^{2}+ \square-\square-3 \\ (f \cdot g)(x) &=6 x^{2}+7x-3 \end{aligned} $$

Step-by-Step Solution

Answer

The product \((f \cdot g)(x)\) is \(6x^2 + 7x - 3\).

Step 1: Define the Operation

We are asked to find \((f \cdot g)(x)\). This represents the product of two functions \(f(x)\) and \(g(x)\).

Step 2: Write Out the Functions

The functions given are \(f(x) = 3x - 1\) and \(g(x) = 2x + 3\).

Step 3: Set Up the Product of Functions

We substitute \(f(x) = 3x - 1\) and \(g(x) = 2x + 3\) into the expression for multiplication: \[(f \cdot g)(x) = f(x) \cdot g(x) = (3x - 1)(2x + 3)\].

Step 4: Apply the Distributive Property

Multiply each term in \((3x - 1)\) by each term in \((2x + 3)\): \[3x \cdot 2x + 3x \cdot 3 - 1 \cdot 2x - 1 \cdot 3\].

Step 5: Execute the Multiplication

Perform the multiplication for each pair of terms: - \(3x \cdot 2x = 6x^2\), - \(3x \cdot 3 = 9x\), - \(-1 \cdot 2x = -2x\), - \(-1 \cdot 3 = -3\).

Step 6: Combine Like Terms

Add the results: \[6x^2 + 9x - 2x - 3\]. Combine the \(x\) terms: \(9x - 2x = 7x\).

Step 7: Present the Final Expression

The simplified product of the functions is: \[(f \cdot g)(x) = 6x^2 + 7x - 3\].

Key Concepts

Composite Functions

When we refer to composite functions in mathematics, we mean a situation where one function is applied to the results of another. However, what we usually describe here in this example is the multiplication of functions, rather than strict composition. In function multiplication, such as finding \((f \cdot g)(x)\), we are multiplying the outputs of \(f(x)\) and \(g(x)\). This differs from composite functions where one function's whole expression is inserted into another.

• For example, for function multiplication, if \(f(x) = 3x - 1\) and \(g(x) = 2x + 3\), then \((f \cdot g)(x) = (3x - 1)(2x + 3)\).
• On the other hand, if we had composite functions, \(f(g(x))\) and \(g(f(x))\), would involve plugging the entire equation of one into the other, which is not required in this instance.

Understanding these distinctions is important as it ensures correct application of mathematical operations and problem-solving.

Distributive Property

The distributive property in algebra is an essential tool used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each addend separately by the number and then adding the products.
In the case of multiplying the functions \((3x - 1)(2x + 3)\), each term in the first parenthesis is multiplied by each term in the second parenthesis:

• \(3x \cdot 2x\), resulting in \(6x^2\),
• \(3x \cdot 3\), resulting in \(9x\),
• \(-1 \cdot 2x\), resulting in \(-2x\),
• \(-1 \cdot 3\), resulting in \(-3\).

This is a direct application of the distributive property, allowing us to transform the product of two binomials into a set of terms that we can further simplify.

Combining Like Terms

Combining like terms is a crucial step in simplifying polynomial expressions. It involves consolidating terms that have the same variable raised to the same power. In the expression obtained from multiplying the functions, such as \(6x^2 + 9x - 2x - 3\), notice the terms \(9x\) and \(-2x\) are 'like' because they both contain the variable \(x\) raised to the first power.
By combining these like terms, \(9x - 2x\), we simplify the expression to \(7x\). This step not only simplifies the expression but also makes it easier to understand and compute.

• Identifying like terms efficiently groups expressions.
• Simplifying complex expressions ensures clarity in solutions.

Working with polynomials often involves several rounds of combining like terms, reflecting the expression more concisely and enabling clearer communication of mathematical ideas.