Problem 14 - 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. $$(f g)(-3)$$

Step-by-Step Solution

Answer

(f g)(-3) = 0.

Step 1: Understand the Composition

The problem asks for the composition \((fg)(x)\). This implies we need to multiply the functions \(f(x)\) and \(g(x)\), resulting in a new function \((fg)(x) = f(x) \, g(x)\).

Step 2: Write the Expression for (fg)(x)

The expression \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\). To find \((fg)(x)\), multiply these two functions: \((fg)(x) = (x^2 + 3x)(2x - 1)\).

Step 3: Expand the Product

To expand \((x^2 + 3x)(2x - 1)\), use the distributive method: \((x^2 + 3x)(2x - 1) = x^2(2x - 1) + 3x(2x - 1)\). Then, distribute: \[x^2 \cdot 2x - x^2 \cdot 1 + 3x \cdot 2x - 3x \cdot 1\].

Step 4: Simplify the Expression

Calculate each term: \(2x^3 - x^2 + 6x^2 - 3x\). Add and combine like terms: \(2x^3 + 5x^2 - 3x\). So, \((fg)(x) = 2x^3 + 5x^2 - 3x\).

Step 5: Evaluate at x = -3

Substitute \(-3\) into the expanded function \((fg)(x) = 2x^3 + 5x^2 - 3x\): \((fg)(-3) = 2(-3)^3 + 5(-3)^2 - 3(-3)\).

Step 6: Calculate Each Term

First, calculate \(2(-3)^3 = 2(-27) = -54\). Next, \(5(-3)^2 = 5(9) = 45\). Finally, \(-3(-3) = 9\).

Step 7: Final Calculation for (fg)(-3)

Add the calculated terms: \(-54 + 45 + 9\). Evaluating these gives \(-54 + 45 + 9 = 0\). Thus, \((fg)(-3) = 0\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions built from variables and constants using operations such as addition, subtraction, multiplication, division, and exponents. They are essential in various areas of mathematics and form the backbone of algebra. In the given exercise, the algebraic functions are defined as \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \). These functions represent equations where each input \( x \) is mapped to an output value calculated using the formula provided in the function.
Understanding how to manipulate and work with these functions, such as combining them through multiplication or composition, is crucial. It helps us explore dynamics in relationships that arise from these algebraic structures. By mastering these concepts, students can solve more complex equations and understand the relationships between different variables.

Polynomial Multiplication

Polynomial multiplication involves expanding expressions constructed of polynomials, which are algebraic functions with multiple terms. Each term comprises a coefficient multiplied by a variable raised to a power. In this exercise, the aim is to multiply two polynomials \( (x^2 + 3x)(2x - 1) \).
Using the distributive property, also known as the algebraic expansion or FOIL method (First, Outer, Inner, Last), we multiply each term in the first polynomial by each term in the second polynomial:

• \(x^2 \cdot 2x = 2x^3\)
• \(x^2 \cdot (-1) = -x^2\)
• \(3x \cdot 2x = 6x^2\)
• \(3x \cdot (-1) = -3x\)

After multiplying, we compile all the resulting terms into a single expression and combine similar, or like, terms. This results in the newly formed polynomial \( 2x^3 + 5x^2 - 3x \). Multiplying polynomials amplifies one's understanding of variables and coefficients, thus enhancing problem-solving abilities in algebra.

Evaluating Functions

Evaluating functions involves finding the value of a function given a specific input. This process often follows expanding or simplifying a function first, as seen in the step-by-step solution.In our exercise, after multiplying the polynomials, we need to evaluate the composite function \((fg)(x) = 2x^3 + 5x^2 - 3x\) at \(x = -3\).
This involves substituting \(-3\) into the simplified expression:

• \( 2(-3)^3 = 2(-27) = -54 \)
• \( 5(-3)^2 = 5(9) = 45 \)
• \( -3(-3) = 9 \)

The final step is to add the results of these calculations: \(-54 + 45 + 9\), which equals \(0\). Hence, \((fg)(-3) = 0\). Learning how to evaluate functions enables students to determine specific outputs, adding practical application to theoretical concepts.