Problem 10 - Algebra Form and Function

Question

Are the two functions the same function? $$ f(x)=2(x+1)(x-3) \text { and } g(x)=x^{2}-2 x-3 $$

Step-by-Step Solution

Answer

Answer: No, the functions f(x) and g(x) are not the same. After simplifying, we have f(x) = 2x^2 - 4x - 6 and g(x) = x^2 - 2x - 3, which are different expressions.

Step 1: Simplify f(x)

We will first simplify the function f(x). Using the given expression: $$ f(x) = 2(x+1)(x-3) $$ Now we will use the distributive property to expand the brackets: $$ f(x) = 2(x^2 - 3x + x - 3) $$ Now we will combine like terms: $$ f(x) = 2(x^2 - 2x - 3) $$ Finally, we will distribute the 2 across all terms inside the brackets: $$ f(x) = 2x^2 - 4x - 6 $$

Step 2: Simplify g(x)

The function g(x) is given by: $$ g(x) = x^2 - 2x - 3 $$ As no further simplification is needed, we can directly compare this function with the simplified f(x).

Step 3: Compare the functions

Now that we have simplified both functions, we can compare their expressions: $$ f(x) = 2x^2 - 4x - 6 $$ $$ g(x) = x^2 - 2x - 3 $$ As we can see, their expressions are not the same. Therefore, f(x) and g(x) are not the same functions.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole-number powers, multiplied by coefficients. These functions are among the most fundamental types of mathematical expressions and often look like combinations of terms such as ax, ax^2, and so on. The function is defined based on the degree, which is the highest power of the variable in the expression. For example, in the expression \(g(x) = x^2 - 2x - 3\), the degree is two, making it a quadratic polynomial.

Key characteristics of polynomial functions include:

• They can have one or more terms.
• Terms are composed of a coefficient and a variable raised to a power.
• The degree of the polynomial is determined by the highest exponent in the function.
• Polynomials are continuous and smooth graphs, which makes them widely used in modeling real-world processes.

Understanding polynomial functions is crucial because they form the basis for more advanced mathematical concepts and applications.

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions. It involves distributing a single term across terms enclosed in parentheses, effectively multiplying each term inside the parentheses by the term outside. This is expressed mathematically as \(a(b + c) = ab + ac\).

In the exercise, this property helps expand the function \(f(x) = 2(x+1)(x-3)\). Here's how it works:

• First, expand \((x+1)(x-3)\) to get \(x^2 - 3x + x - 3\).
• Combine like terms to simplify into \(x^2 - 2x - 3\).
• Use the distributive property \(2(x^2 - 2x - 3)\) to get \(2x^2 - 4x - 6\).

The distributive property is a powerful tool that simplifies complex expressions and is frequently used in algebraic manipulations and solving equations.

Simplification Steps

Simplification is the process of reducing expressions to their most basic forms, making them easier to understand and compare. This is done through combining like terms, using the distributive property, and reorganizing expressions efficiently.

Using the simplification steps in the exercise:

• Initial Expression: Start with \(f(x) = 2(x+1)(x-3)\).
• Apply Expansion: Use distributive property to expand \((x+1)(x-3)\).
• Combine Like Terms: Combine terms to get \(x^2 - 2x - 3\).
• Final Simplification: Multiply each term by 2 to finalize \(f(x) = 2x^2 - 4x - 6\).

These steps show how an expression can be effectively reduced to a simpler form, enabling straightforward comparison or further mathematical operations.