Problem 2 - Algebra 2

Question

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ g(x)-f(x) $$

Step-by-Step Solution

Answer

The simplified expression for \(g(x)-f(x)\) is \(x^{2}-3x-5\).

Step 1: Write out the functions

The function \(f(x)=3x+5\) and \(g(x)=x^2\). We are asked to compute \(g(x)-f(x)\), which means subtracting the entire \(f(x)\) expression from \(g(x)\).

Step 2: Subtract f(x) from g(x)

To find the difference \(g(x)-f(x)\), replace \(f(x)\) and \(g(x)\) in this expression with their respective function expressions. This gives us \(x^2-(3x+5)\).

Step 3: Simplify the expression

We simplify this expression to \(x^{2}-3x-5\). This is done by distributing the negative sign across the terms \(3x\) and \(5\) in the subtraction \((3x+5)\) and then combining like terms.

Key Concepts

Function Operations

When working with algebraic functions, like in the given exercise, understanding how to perform function operations is crucial. Functions are like machines where you input a value and get a specific output. Function operations involve applying basic arithmetic operations, such as addition, subtraction, multiplication, and division, to these functions.

• In this case, we are dealing with the subtraction of two functions. Consider the functions \(f(x) = 3x + 5\) and \(g(x) = x^2\).
• The operation \(g(x) - f(x)\) asks us to subtract \(f(x)\) from \(g(x)\).
• This means we evaluate both functions separately and then subtract the resulting expressions to find a new function that represents their difference.

When performing such operations, keeping track of positive and negative signs, as well as like terms, is important to avoid errors in calculations.

Polynomial Subtraction

Polynomial subtraction is a specific type of function operation that entails subtracting one polynomial from another. Here, polynomials are expressions consisting of variables and coefficients, usually with varying powers. In the example of \(g(x) - f(x)\), we subtract the polynomial \(3x + 5\) from \(x^2\). The process for polynomial subtraction involves a few key steps:

• First, write out the full expression for both functions involved in the subtraction.
• Next, distribute the subtraction across the full polynomial, which means applying the negative sign to each term in the polynomial being subtracted.
• In this exercise, we have \(x^2 - (3x + 5)\), which becomes \(x^2 - 3x - 5\) after distributing the negative sign.

Understanding how to properly apply this operation can help simplify complex problems and is foundational for working with algebraic expressions.

Simplifying Expressions

Once the polynomial subtraction is completed, the next essential step is simplifying the resulting expression. Simplification involves combining like terms and ensuring the expression is in its simplest form for interpretation or further manipulation. Here is how simplification works:

• Identify like terms: In our example, after subtraction, we have \(x^2 - 3x - 5\). Each term here is distinct, so there are no like terms to combine.
• Ensure all operations have been completed correctly: Double-checking negative signs and ensuring all terms have been accounted for ensures accuracy.
• The expression \(x^2 - 3x - 5\) cannot be simplified further as there are no like terms and no arithmetic operations left to perform.

Simplifying expressions allows you to present solutions in their clearest and most usable form, essential for both calculating results and communicating solutions effectively.