Problem 13 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f g)(4)$$

Step-by-Step Solution

Answer

(fg)(4) = 196.

Step 1: Understand the Operation (fg)(x)

The expression \((fg)(x)\) represents the product of the functions \(f(x)\) and \(g(x)\). So, \((fg)(x) = f(x) \, g(x)\). This means we need to multiply the two functions together.

Step 2: Substitute Functions

Given \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\), substitute these into the expression for \((fg)(x)\): \[(fg)(x) = (x^2 + 3x)(2x - 1)\]

Step 3: Distribute the Terms

Use the distributive property to expand the products: \[x^2(2x - 1) + 3x(2x - 1)\]This becomes: \[2x^3 - x^2 + 6x^2 - 3x\]

Step 4: Combine Like Terms

Combine the like terms in the expression: \[2x^3 + 5x^2 - 3x\] This is the expression for \((fg)(x)\).

Step 5: Evaluate (fg)(4)

Substitute \(x = 4\) into the combined expression: \[2(4)^3 + 5(4)^2 - 3(4)\]Evaluate each term: \[2 \cdot 64 + 5 \cdot 16 - 12\]\\[128 + 80 - 12\] \= 196.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to non-negative integer exponents. For instance, in the given exercise, the function \( f(x) = x^2 + 3x \) is a polynomial function. Here, you notice two terms: \( x^2 \) and \( 3x \). The first term, \( x^2 \), is a quadratic term, and the second term, \( 3x \), is linear. Polynomial functions are the core elements in algebra and are often used in equations to model a variety of real-world situations.

Polynomials can be combined with other functions using operations such as addition, subtraction, multiplication, and division.

• Example of polynomial: \( 4x^3 + 5x^2 + x + 6 \)
• Terms in the polynomial are separated by \(+\) or \(-\).

Understanding these basics helps in manipulating and operating polynomial expressions effectively.

Distributive Property

The distributive property is one of the fundamental rules when dealing with algebraic expressions, especially useful in multiplying polynomials. The property states that a term multiplied by a sum of terms can be rewritten as the sum of products. In mathematical terms, it translates to: \( a(b+c) = ab + ac \).

In our exercise, the distributive property is used to multiply the expressions from functions \( f(x) \) and \( g(x) \). Given \( (x^2 + 3x)(2x - 1) \), you apply the distributive property to expand:

• First, distribute \( x^2 \) across \( (2x - 1) \): \( x^2 \cdot 2x - x^2 \cdot 1 \).
• Next, distribute \( 3x \) across \( (2x - 1) \): \( 3x \cdot 2x - 3x \cdot 1 \).

This results in a new expression: \( 2x^3 - x^2 + 6x^2 - 3x \). The distributive property is pivotal in ensuring every term is accurately considered in the multiplication process.

Combining Like Terms

Combining like terms is a simplification process where similar terms in an expression are joined together to make calculations easier and the polynomial clearer. Like terms are terms that have the same variable raised to the same power. For example, \( 5x^2 \) and \( -x^2 \) are like terms because they both involve the variable \( x \) raised to the power of 2.

In our exercise, after applying the distributive property, you get the expression: \( 2x^3 - x^2 + 6x^2 - 3x \). Here's how you combine like terms:

• Identify the like terms: \( -x^2 \) and \( 6x^2 \).
• Combine them: \( -x^2 + 6x^2 = 5x^2 \).

This results in a simplified expression: \( 2x^3 + 5x^2 - 3x \). Mastering this process is important as it decreases complexity and makes evaluating expressions more manageable.