Problem 2 - A Graphical Approach to Precalculus with Limits

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. $$I$$ $$(f-g)(x)$$ $$\mathbf{II}$$ $$A.\quad4 x^{2}-20 x+25$$ $$B.\quad x^{2}-2 x+5$$ $$C.\quad 2 x^{2}-5$$ $$D.\quad \frac{x^{2}}{2 x-5}$$ $$E. \quad x^{2}+2 x-5$$ $$F. \quad 2 x^{3}-5 x^{2}$$

Step-by-Step Solution

Answer

(f-g)(x) matches with expression B: \(x^2 - 2x + 5\).

Step 1: Understand (f-g)(x)

The expression \((f-g)(x)\) represents the subtraction of the function \(g(x)\) from \(f(x)\). To find this, we need to use the formula \((f-g)(x) = f(x) - g(x)\).

Step 2: Substitute f(x) and g(x) into (f-g)(x)

Substitute \(f(x) = x^2\) and \(g(x) = 2x - 5\) into \((f-g)(x)\). This gives \((f-g)(x) = x^2 - (2x - 5)\).

Step 3: Simplify the expression

First expand the subtraction: \(x^2 - (2x - 5) = x^2 - 2x + 5\). Simplified, this gives the expression \(x^2 - 2x + 5\).

Step 4: Find the correct match in Group II

Compare the result \(x^2 - 2x + 5\) to the expressions given in Group II: A, B, C, D, E, and F. The expression matches choice B: \(x^2 - 2x + 5\).

Key Concepts

Polynomial functions

A polynomial function is a mathematical expression that involves a sum of powers and coefficients, with an emphasis on whole number exponents. These functions are foundational in algebra because they model a wide array of real-world situations and maintain a friendly structure for manipulation and simplification. In our exercise, we encounter polynomial functions in the form of two expressions: \(f(x) = x^2\) and \(g(x) = 2x - 5\). Each of these can be broken down into simpler polynomial expressions.

• \(f(x) = x^2\) is a basic polynomial, representing a quadratic function primarily due to its highest exponent of 2. It forms a parabolic curve when graphed.
• \(g(x) = 2x - 5\) is a linear polynomial, characterized by its first-degree expression, forming a straight line graph.

These polynomial functions can be combined, manipulated, and simplified, offering a basis for understanding more complex algebraic concepts.

Expression simplification

Simplifying algebraic expressions is a critical skill in algebra that involves reducing expressions to their simplest form. This often requires combining like terms, removing parentheses, and making the expression as concise and clear as possible.
In our step-by-step solution, the simplification process begins with understanding the operation \((f-g)(x)\). This requires subtracting one polynomial from another, specifically \(g(x)\) from \(f(x)\).

• We substitute the given values, \(f(x) = x^2\) and \(g(x) = 2x - 5\), into the expression \((f-g)(x) = x^2 - (2x - 5)\).
• Then, we distribute the negative sign across the terms inside the parentheses, leading to \(x^2 - 2x + 5\).

The result is a simplified expression that maintains the integrity of the original function values while removing extraneous complexity.
This simplification is crucial in choosing the right match from the given options.

Algebraic expressions

Algebraic expressions form the core of algebra by representing numbers and variables in compact forms. The operations within algebraic expressions include addition, subtraction, multiplication, and division of numbers and variables, integrated via mathematical symbols and operators.
In this exercise, the task was to work with algebraic expressions derived from polynomial functions. The chief skill involved was function subtraction, specifically subtracting the function \(g(x)\) from \(f(x)\), producing the new expression \(x^2 - 2x + 5\).

• Understanding how to substitute and rearrange these expressions lays the groundwork for solving more complicated algebra problems.
• After substitution, the focus was on accurately managing the arithmetic operations to simplify the expression, highlighting the algebraic expression's adaptability and mathematical beauty.

This introduction to algebraic expressions and their manipulation sets a foundation for understanding deeper algebraic principles, from polynomials to rational expressions and beyond.