Problem 7 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The result of \(f + g\), i.e., \( (f + g)(x)\) is \(x^{2} + 3x + 5\).

Step 1: Determine the Expressions for the Functions

The expressions for the functions are given as \(f(x) = 3x + 5\) and \(g(x) = x^{2}\).

Step 2: Perform Function Operation

The function operation is addition or \(f + g\). To perform this operation, add the expressions for \(f(x)\) and \(g(x)\) together. The expression for \(f(x) + g(x)\) is \(3x + 5 + x^{2}\).

Step 3: Arranging the Result Ascendingly

It's conventional to write this in ascending order of power as \(x^{2} + 3x + 5\).

Key Concepts

Polynomial Functions

Polynomial functions are foundational elements in algebra that consist of variables, coefficients, and exponents. They can take various forms, with the degree of the polynomial being determined by the highest exponent of its variable. For instance, in our exercise, function \(g(x) = x^2\) is a simple quadratic polynomial. Quadratic indicates that the highest exponent is 2. Polynomials can be as simple as a constant or linear function \((e.g., \, f(x) = 3x + 5)\), or more complex like cubic, quartic, and beyond.

Understanding how to identify and manipulate polynomial functions is essential in solving algebraic expressions. When dealing with polynomial functions, the key is to be able to recognize the degree and organize the terms by descending or ascending order based on their exponents. This allows you to apply operations such as addition, subtraction, and even multiplication easily.

Function Addition

Function addition is a process by which two or more functions are combined to form a new function. Essentially, you're adding together similar functions \(\(f(x)\) and \(g(x)\) in this case\). Adding functions involves summing their respective outputs for each input value. In mathematical terms, if \(f(x) = 3x + 5\) and \(g(x) = x^2\), then \((f+g)(x) = f(x) + g(x)\) is computed as \(3x + 5 + x^2\).

To correctly perform function addition, follow these steps:

• Ensure both functions are expressed in similar forms – usually polynomials for ease of computation.
• Identify like terms, such as terms with the same variable to the same power, and combine them.
• Write the resulting expression clearly, often in standard form (ascending or descending power order).

By practicing function addition, you enhance your ability to manipulate and combine different types of algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They're the basic building blocks of algebra and can vary in complexity. In the given exercise, the expressions \(3x + 5\) and \(x^2\) are algebraic since they contain numbers and variables combined by operations like addition and multiplication.

When working with algebraic expressions, it is critical to:

• Identify and correctly interpret coefficients (e.g., 3 in \(3x+5\)), variables (e.g., \(x\)), and exponents (e.g., 2 in \(x^2\)).
• Understand that like terms must have the same variable and exponent to be combined.
• Explore operations involving expressions, such as addition, subtraction, expansion, and simplification.

Mastering algebraic expressions aids in performing complex mathematical tasks, paving the way for understanding advanced topics like calculus and differential equations.