Problem 73 - Algebra 2

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+1 .\) Perform each function operation and use the properties of real numbers to justify each step in simplifying your answer. $$ (f+g)(x) $$

Step-by-Step Solution

Answer

The sum of the functions \(f(x)\) and \(g(x)\) is \(x^{2} + 3x - 1\).

Step 1: Express the Sum of the Functions

The sum of two functions, say \(f(x)\) and \(g(x)\), is defined as \( (f+g)(x) = f(x) + g(x) \). Replace \(f(x)\) with \(3x - 2\), and \(g(x)\) with \(x^{2}+1\), to get \( (f+g)(x) = (3x - 2) + (x^{2}+1)\).

Step 2: Simplify the Sum

The expression for the sum of the functions is \( (3x - 2) + (x^{2}+1) \). Use the commutative property of addition (which states that changing the order of addends does not change the sum) to rearrange the terms: \( (f + g) (x) = x^{2} + 3x - 2 + 1\).

Step 3: Combine Like Terms

To finally simplify this sum, combine the like terms: -2 + 1. Therefore, \( (f + g) (x) = x^{2} + 3x - 1\). This is the simplified form of the sum of \(f(x)\) and \(g(x)\).

Key Concepts

Understanding Functions

At its core, a function is like a machine that takes an input, processes it, and then gives an output. In the world of mathematics, a function is a specific relation where each input (often represented as \(x\)) has exactly one output (often represented as \(f(x)\) or \(g(x)\)).
The function \(f(x) = 3x - 2\) processes the input \(x\) by multiplying it by 3 and then subtracting 2. On the other hand, \(g(x) = x^2 + 1\) squares the input and then adds 1.
Understanding these basic operations is crucial for mastering different function operations such as addition, subtraction, multiplication, and division of functions.

• Function addition is simply defined as \((f+g)(x) = f(x) + g(x)\).
• Function subtraction, multiplication, and division have similar definitions, allowing us to combine or transform functions in numerous ways.

Learning to manipulate functions through these operations is fundamental in algebra and calculus. It helps describe complex relationships between variables in various fields, from physics to economics.

Exploring Real Numbers

Real numbers are a set of numbers containing both rational and irrational numbers. This means numbers that can be fractions, whole numbers, or decimals that never end. Essentially, real numbers encompass everything except imaginary numbers.
In our function operations, we deal with real numbers all the time. When adding the functions \(f(x) = 3x - 2\) and \(g(x) = x^2 + 1\), all the terms and coefficients like 3, -2, and 1 fall under real numbers. Values of \(x\) can also be any real numbers.
Using the properties of real numbers, such as the commutative property (changing the order of addends or factors does not change their sum or product), ensures that our calculations can be done in a flexible manner. This property is used when rearranging terms to simplify the expression of function sums or differences. Similarly, combining like terms, a key step in simplification, relies on the real number properties to yield a clean and precise expression for function operations.

The Art of Simplification

Simplification is all about making expressions easier to work with by reducing them to their most concise form. The process involves several key steps.

1. **Rearranging Terms:** Use properties like commutativity to change the order of terms for easier manipulation.2. **Combining Like Terms:** Identify and add together terms with the same variable and power. For example, in the expression \((f+g)(x) = x^2 + 3x - 2 + 1\), the terms \(-2\) and \(1\) are combined to give \(-1\), leading to the simplified form \(x^2 + 3x - 1\).

Proper simplification not only makes the expressions neat but also reveals insights that are otherwise buried under complexity. For instance, expressing functions in their simplest form allows for easier graphing or solving differential equations.
So, simplification remains an essential skill in mathematical problem-solving, requiring keen attention to detail and a deep understanding of mathematical properties.