Problem 1 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The sum of the functions \(f(x)\) and \(g(x)\) is \(f(x) + g(x) = x^2 + 3x + 5\).

Step 1: Identify the given functions

The given functions are \(f(x)=3x+5\) and \(g(x)=x^2\).

Step 2: Perform the Addition of the Functions

The sum of the given functions \(f(x)\) and \(g(x)\) is found by adding the expressions for \(f(x)\) and \(g(x)\). This gives \(f(x) + g(x) = (3x + 5) + x^2\). It is important to note that the terms cannot be combined because they are not like terms.

Step 3: Simplify the Expression

To simplify the expression, just rewrite it by ordering the terms according to their degrees in decreasing order which gives \(f(x) + g(x) = x^2 + 3x + 5\).

Key Concepts

Function Operations

When dealing with algebraic functions such as polynomials, there are multiple operations that you can apply. Common function operations include addition, subtraction, multiplication, and division. In our specific example, we focus on addition. Adding functions is straightforward: you sum their outputs.

For example, given functions \( f(x) = 3x + 5 \) and \( g(x) = x^2 \), adding them involves combining their expressions:

• Write down both expressions: \( f(x) + g(x) \)
• Substitute with the expressions: \( (3x + 5) + x^2 \)

This shows how you directly combine the outputs of the two functions by linking their equations with a plus sign.

It's important to maintain each step in your function operation clear to ensure no mistakes occur, such as wrongly combining unlike terms.

Polynomial Expressions

Polynomials are algebraic expressions that consist of variables and coefficients linked with operations of addition, subtraction, and multiplication. For instance, our functions \( f(x) = 3x + 5 \) and \( g(x) = x^2 \) are polynomial expressions.

Every polynomial is made up of terms that consist of a coefficient and a variable raised to an exponent. Consider \( g(x) \); it has a single term, \( x^2 \), where the coefficient is 1 and the variable \( x \) is raised to an exponent of 2.

Polynomials are widely used because they can form the foundation of many algebraic computations, and they follow strict rules that simplify the process of function operations.

Like Terms

"Like terms" in algebra are terms within an expression that have identical variable parts raised to the same powers. Recognizing and combining like terms is crucial in simplifying expressions.

• For instance, in \( f(x) + g(x) = (3x + 5) + x^2 \), the term \( 3x \) cannot be combined with \( x^2 \) because they have different exponents.
• The constant term \( 5 \) doesn't have a corresponding like term in \( g(x) \).

Understanding this concept ensures clarity and correctness. Mixing unlike terms can easily lead to mistakes in calculations and misinterpretation of an expression.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible while keeping their value unchanged. Once you've combined functions and identified like terms, you should always try to simplify the expression.

In our equation \( f(x) + g(x) = x^2 + 3x + 5 \), simplification involves arranging terms by their degree's order: starting from the highest exponent to the lowest.

Although this expression \( x^2 + 3x + 5 \) cannot be further combined, organizing terms demonstrates a neat, approachable layout that can be easily interpreted or used for further operations or graphing. This technique helps in reducing complexity and allows easier analysis and application of the equation.