Problem 2 - 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

(f-g)(x) = x^2 - 2x + 5, which is Group II option B.

Step 1: Define the Functions

Given the functions: \( f(x) = x^2 \) and \( g(x) = 2x - 5 \). These will be used to find the expression for \( (f-g)(x) \).

Step 2: Subtract the Functions

Subtract \( g(x) \) from \( f(x) \). This will give \( (f-g)(x) = f(x) - g(x) \).

Step 3: Apply the Subtraction

Substitute the given functions into the expression: \( (f-g)(x) = x^2 - (2x - 5) \).

Step 4: Simplify the Expression

Distribute the negative sign and combine like terms: \( (f-g)(x) = x^2 - 2x + 5 \).

Step 5: Identify the Matching Expression

Look for the simplified expression \( x^2 - 2x + 5 \) in Group II. It matches with option B.

Key Concepts

Function Operations

Algebraic functions like those in our exercise, work similarly to numbers. You can add, subtract, multiply, and divide them. This is known as function operations. How can you perform these operations? By directly applying them to the defined functions using algebraic manipulations.

Let's think about function subtraction. It involves taking one function and subtracting another function from it. In this exercise, we have two functions:

• The first is: \(f(x) = x^2\)
• The second is: \(g(x) = 2x - 5\)

For subtraction, the notation \((f-g)(x)\) stands for \(f(x) - g(x)\). Substitute the expressions for \(f\) and \(g\) and calculate the result.

Polynomial Subtraction

Polynomial subtraction can often sound intimidating, but it is more straightforward than you might think. We start by writing both functions in standard polynomial form, as seen in the exercise:

• \(f(x) = x^2\)
• \(g(x) = 2x - 5\)

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\): \[(f-g)(x) = x^2 - (2x - 5)\]Be careful with the negative sign. Subtracting means distributing the negative sign across all terms in \(g(x)\).

• Start with \(x^2\) (no change here)
• Subtract \(2x\)
• Add 5 (since \(-(-5) = +5\))

This results in a new expression: \(x^2 - 2x + 5\).

Expression Simplification

Simplifying expressions is a crucial step in algebra. It helps make expressions easier to handle and understand. Once polynomial subtraction was performed in the given exercise, we reached this expression:

• \(x^2 - 2x + 5\)

To simplify, check for like terms. These are terms with the same variable raised to the same power, which can be combined. In our result, the terms \(-2x\) and \(5\) cannot be simplified further because they are not like terms.Simplification also involves distributing negative signs and arranging terms in standard form, which is descending order based on the power of the variable. As you can see, the simplified expression remains \(x^2 - 2x + 5\) and correctly corresponds to option B in Group II.