Problem 45 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The short answer is \(x^2 - x + 7\). This is obtained by adding the functions \(f(x)\) and \(g(x)\) together and simplifying the result.

Step 1: Define the Functions

The problem provides two functions. The first function is \(f(x) = 2x + 5\) and the second function is \(g(x) = x^2 - 3x + 2.\)

Step 2: Perform the Addition Operation

To add those functions together, the terms with the same exponents are added. So we take \(f(x) + g(x)\) which will be \((2x + 5) + (x^2 - 3x + 2)\).

Step 3: Simplify the Expression

The expression simplifies to \(x^2 - x + 7\) because you add like terms, i.e., the x terms (2x - 3x) and the constant terms (5 + 2).

Key Concepts

Function Operations

Function operations involve combining and manipulating functions in various ways. One common operation is addition, where you combine two functions into one. For example, given functions \(f(x) = 2x + 5\) and \(g(x) = x^2 - 3x + 2\), performing \(f(x) + g(x)\) means you're adding these two expressions.
This operation involves:

• Aligning the terms based on their exponents, such as constants, linear terms, and quadratic terms.
• Then, you'll add coefficients of terms with the same degree, noting that terms that do not have matching counterparts remain as is.

Think of it like combining ingredients in a recipe; you have to carefully consider each part's contribution to the whole. After performing the addition, the expression becomes a single, cohesive polynomial that combines effects from both functions.

Polynomial Simplification

Simplifying polynomials is a process of making the expression as compact and efficient as possible by reorganizing and combining its terms. Suppose you have the expression \(f(x) + g(x) = (2x + 5) + (x^2 - 3x + 2)\).

In simplification, you break down each part:

• First, group similar terms. Here, \(x^2\) stands alone as a quadratic term.
• Next, combine the linear \(x\) terms: \(2x\) and \(-3x\) simplify to \(-x\).
• Finally, add constant terms: \(5 + 2\) equals \(7\).

After simplification, the whole polynomial is just \(x^2 - x + 7\). This process eliminates any redundancy and makes the function easier to interpret and use.

Combining Like Terms

Combining like terms is crucial in simplifying algebraic expressions. This practice focuses on grouping terms with the same powers or exponents and performing arithmetic operations on them. In the given expression, \((2x + 5) + (x^2 - 3x + 2)\), you will:

• Identify terms by their degree. For instance, all terms that include \(x\), like \(2x\) and \(-3x\), are grouped together.
• Combine those terms by adding or subtracting their coefficients. So, \(2x - 3x\) becomes \(-x\).
• Apply the same method to remaining constants and higher degree terms. Here, the constants \(5\) and \(2\) become \(7\).

This method not only simplifies the process but also clarifies how the polynomial behaves by summarizing it into manageable parts. It's a fundamental skill in algebra, aiding in everything from solving equations to graphing functions.