Problem 31 - Elementary Algebra

Question

Given \(f(x)=16 \times 2-9 x+5\) and \(g(x)=x 2+3 x-104 x 2+5 x-6,\) calculate \((f \cdot g)(x)\) and state the restrictions.

Step-by-Step Solution

Answer

The product is \(927x^3 - 3883x^2 + 350x - 222\). No restrictions on domain.

Step 1: Simplify Functions

First, simplify the given functions. For \(f(x)\), simplify: \[ f(x) = 16 \times 2 - 9x + 5 \]which simplifies to: \[ f(x) = 32 + 5 - 9x = 37 - 9x \]For \(g(x)\), simplify: \[ g(x) = x^2 + 3x - 104x^2 + 5x - 6 \]Combine like terms:\[ g(x) = (1 - 104)x^2 + (3 + 5)x - 6 = -103x^2 + 8x - 6 \]

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

To find \((f \cdot g)(x)\), multiply the simplified forms of \(f(x)\) and \(g(x)\): \[ (f \cdot g)(x) = (37 - 9x)(-103x^2 + 8x - 6) \] Distribute each term:1. \(37 \times -103x^2 = -3811x^2\)2. \(37 \times 8x = 296x\)3. \(37 \times -6 = -222\)4. \(-9x \times -103x^2 = 927x^3\)5. \(-9x \times 8x = -72x^2\)6. \(-9x \times -6 = 54x\)Combine all terms:\[ (f \cdot g)(x) = 927x^3 + (-3811x^2 - 72x^2) + (296x + 54x) - 222 \]Simplify:\[ (f \cdot g)(x) = 927x^3 - 3883x^2 + 350x - 222 \]

Step 3: Determine Restrictions

Since there are no denominators or square roots involved, there are no domain restrictions based on those conditions. Therefore, the domain of \((f \cdot g)(x)\) is \(\mathbb{R}\).

Key Concepts

Function Simplification

Simplifying functions involves expressing the given algebraic expressions in their most reduced form. Simplification makes mathematical expressions easier to handle and more understandable by eliminating unnecessary components.
To simplify a function, follow these steps:

• Combine like terms, which are terms that have the same variable and power.
• Perform any arithmetic operations such as addition or subtraction with coefficients.

In our exercise, we simplified functions \[ f(x) = 16 \times 2 - 9x + 5 \] to \[ f(x) = 37 - 9x \]. Similarly, for\[ g(x) = x^2 + 3x - 104x^2 + 5x - 6 \], we combined like terms resulting in\[ g(x) = -103x^2 + 8x - 6 \].
These reduced expressions are tidier and easier to work with in further calculations, such as when multiplying the functions together.

Domain Restrictions

Understanding domain restrictions is crucial when dealing with functions. The domain of a function comprises all the possible input values (or 'x' values) that do not cause the function to produce undefined or invalid results.
Common domain restrictions occur in cases:

• Where there are fractions, and the denominator cannot be zero.
• Where there are square roots, and the expression inside the square root must be non-negative (for real number systems).

In our problem, we found that there are no denominators or square roots in \((f \cdot g)(x)\), which meant that every real number is a valid input. Therefore, the domain is \( \mathbb{R} \), indicating no restrictions.

Algebraic Expressions

Algebraic expressions are mathematical statements that include numbers, variables, and operations. Understanding how to manipulate and operate on these expressions is foundational in algebra.
Key points about algebraic operations include:

• Addition and subtraction of like terms: Only terms with the same variable and exponent can be combined.
• Multiplication: When multiplying expressions, distribute each part of one expression through the other.
• Order of operations: Follow the correct sequence when performing computations (parentheses, exponents, multiplication/division, addition/subtraction).

The multiplication of \((f(x) \cdot g(x))\) involved distributing each term of \(f(x)\) to each term of \(g(x)\), and combining like terms in the result. Keeping these principles in mind will help when tackling more complex algebraic operations.