Problem 3 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group II. (Group II) A. \(4 x^{2}-20 x+25\) B. \(x^{2}-2 x+5\) C. \(2 x^{2}-5\) D. \(\frac{x^{2}}{2 x-5}\) E. \(x^{2}+2 x-5\) F. \(2 x^{3}-5 x^{2}\) (Group I) $$(f g)(x)$$

Step-by-Step Solution

Answer

\((fg)(x)\) corresponds to option F: \(2x^3 - 5x^2\).

Step 1: Understand the Function Composition

The expression \((fg)(x)\) represents the product of the functions \(f(x)\) and \(g(x)\). This means we need to multiply the two functions together. Express this as \(f(x) \cdot g(x)\).

Step 2: Substitute the Function Expressions

Substitute \(f(x) = x^2\) and \(g(x) = 2x - 5\) into the expression for \((fg)(x)\):\[(fg)(x) = (x^2)(2x - 5)\]

Step 3: Multiply the Functions

Multiply \(x^2\) by the terms in \(g(x) = 2x - 5\):\[(x^2)(2x - 5) = x^2(2x) - x^2(5)\]

Step 4: Simplify the Expression

Calculate the products:- \(x^2 \cdot 2x = 2x^3\)- \(x^2 \cdot 5 = 5x^2\)Place them together to get the expression:\[2x^3 - 5x^2\]

Step 5: Match with Group II

Compare the simplified function \((fg)(x) = 2x^3 - 5x^2\) to the options in Group II. The correct match is option \(F: 2x^3 - 5x^2\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to positive integer powers and coefficients. These are some of the most common functions found in algebra and calculus. An easy example is a quadratic polynomial:\[f(x) = x^2 + 3x + 2\]Here, this polynomial includes terms with powers of 2 and 1, with the constant term 2. When working with polynomial functions, it’s essential to understand the degree of a polynomial, which is the highest power of its variable. For instance, the degree of \(f(x) = x^2 + 3x + 2\) is 2 because the highest power of the variable \(x\) is 2.
Polynomials can look more complex, such as:\[2x^3 - 4x^2 + 6x - 3\]This polynomial has a degree of 3 due to the term \(2x^3\). Polynomials are foundational in mathematics because they are used for building various forms, analyzing functions, and solving equations.

Multiplying Functions

When you multiply two functions together, you combine them using their algebraic expressions. In essence, the idea is to take each part of one function and multiply it by each part of another function. For instance, given two functions as defined in our example:- \( f(x) = x^2 \)- \( g(x) = 2x - 5 \) To find \((fg)(x)\), follow these steps:

• Substitute the expressions: Replace \(f(x)\) and \(g(x)\) with their definitions:

\[ (fg)(x) = (x^2)(2x - 5) \]Then, apply the distributive property:

• Multiply \(x^2\) by each term in \(g(x)\):
• \(x^2 \cdot 2x = 2x^3\)
• \(x^2 \cdot (-5) = -5x^2\)

Combine those results:
\[2x^3 - 5x^2\]Notice how you add the results to get a combined expression. Understanding how to multiply such functions is critical in manipulating algebraic expressions and solving equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operational symbols, like addition (+), subtraction (-), multiplication (×), and division (÷).
They form the foundation of algebra. An expression can be as simple as a single number or variable, like \(7\) or \(x\), or as complicated as a polynomial like \(3x^4 - 6x^2 + x - 9\).
When working with algebraic expressions, keep these points in mind:

• Terms: Parts of the expression separated by plus or minus signs. For example, in \(3x^2 + 5x - 7\), terms are \(3x^2\), \(5x\), and \(-7\).
  
• Coefficients: Numbers in front of variables that imply multiplication. In \(8x\), 8 is the coefficient.
  
• Variables: Symbols that stand in for unknown values. In our examples, \(x\) is a variable.
  
• Constants: Standalone numbers without accompanying variables. In \(9x + 3\), the constant is 3.
  

Algebraic expressions should be simplified to reduce complexity by combining like terms, factoring, or performing other operations. Clear understanding and manipulation of algebraic expressions are crucial for solving equations and understanding functions in mathematics.