Problem 13 - A Graphical Approach to Precalculus with Limits

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

The result of \((fg)(4)\) is 196.

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

The operation \((fg)(x)\) represents the product of the two functions \(f(x)\) and \(g(x)\). To perform \((fg)(x)\), we need to multiply \(f(x)\) by \(g(x)\), resulting in the expression: \(f(x) \cdot g(x)\).

Step 2: Substitute and Simplify Functions

Substitute the given formulas for \(f(x)\) and \(g(x)\) into the expression. \[f(x)g(x) = (x^2 + 3x)(2x - 1)\]. Simplify this by using the distributive property to multiply each term in \(f(x)\) by each term in \(g(x)\): \[= x^2(2x - 1) + 3x(2x - 1)\].

Step 3: Perform the Multiplication

Distribute the terms: \[x^2(2x) - x^2(1) + 3x(2x) - 3x(1)\]. Simplify these: \[= 2x^3 - x^2 + 6x^2 - 3x\].

Step 4: Simplify the Expression

Combine like terms resulting from the multiplication: \[= 2x^3 + 5x^2 - 3x\].

Step 5: Evaluate the Expression at x = 4

Substitute \(x = 4\) into the simplified expression: \[2(4)^3 + 5(4)^2 - 3(4)\]. Compute the result for each term: \[= 2(64) + 5(16) - 12\].

Step 6: Calculate the Final Result

Calculate each term: \(2(64) = 128\), \(5(16) = 80\), and \(-12\). Combine these results: \[128 + 80 - 12 = 196\].

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental part of algebra, representing mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients. In simple terms, a polynomial function is like a recipe where each ingredient has a numeric multiplier.

Some important characteristics of polynomial functions include:

• Each term in a polynomial is made up of a coefficient (a numerical factor), a variable (like \(x\)), and an exponent that indicates the power to which the variable is raised.
• A polynomial function is generally expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer.
• The highest exponent in the polynomial function signifies the degree of the polynomial. For instance, in the polynomial \(x^2 + 3x\), the degree is 2 because the highest power is \(x^2\).

Polynomials can range from simple (like linear polynomials) to complex (like quartic polynomials) and are used across numerous fields including physics, engineering, and computer science. Understanding their structure is crucial before performing any operations, such as function composition.

Distributive Property

The distributive property is a key algebraic property that allows you to multiply a term by a sum or difference inside parentheses. This property is represented with the formula:

• \(a(b + c) = ab + ac\)

This means that you distribute the multiplication over each term within the parentheses.

In function composition, especially when multiplying polynomial functions, the distributive property plays an essential role:

• Each term in the first polynomial is multiplied by each term in the second polynomial.
• This process ensures that no part of the multiplication is left out, making it possible to simplify expressions and combine like terms afterwards.

Consider the composition \(f(x)g(x) = (x^2 + 3x)(2x - 1)\). Applying the distributive property helps break down the problem into manageable parts:

• First, multiply \(x^2\) by each term in \(g(x)\).
• Next, do the same for the term \(3x\).

This rule is foundational in algebra and essential when working with expressions involving polynomial functions.

Function Evaluation

Function evaluation involves computing the output of a function for specific input values. It is essentially plugging in a number for \(x\) in our function formula.

For example, if you have generated a new function through composition, the next step is often to evaluate it at a particular value of \(x\). This is what was done in the problem with \(x = 4\).

• First, substitute the specified value into the composed function, here \((f g)(x)\).
• Next, solve the expression by performing arithmetic operations like multiplication, addition, and subtraction.

In the example, the simplified expression \(2x^3 + 5x^2 - 3x\) was evaluated at \(x = 4\):

• Calculate \(2(4)^3 = 128\), \(5(4)^2 = 80\), and \(-3(4) = -12\).
• Finally, sum these results: \(128 + 80 - 12 = 196\).

Evaluating functions is a crucial skill because it provides tangible results from algebraic expressions, confirming that mathematical principles are used correctly.