Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

To find \(g(x+1),\) what do you substitute for \(x\) in the function \(g(x)=3 x-2 ?\)

Step-by-Step Solution

Answer

Substituting \(x+1\) into the given function gives \(g(x+1) = 3x + 1\).

Step 1: Identify the given function

The given function is \(g(x) = 3x - 2\).

Step 2: Substitute Identifying terms

The exercise calls for finding \(g(x+1)\), so replace \(x\) in the function with \(x+1\). The substituted function becomes \(g(x+1) = 3*(x+1) - 2\).

Step 3: Simplify the function

Next, the function must be simplified. Use the distributive law to distribute the 3 across the inside of the parentheses. The simplified function then becomes \(g(x+1) = 3x + 3 - 2\). Further simplification gives: \(g(x+1) = 3x + 1\).

Key Concepts

Function Simplification

Function simplification involves reducing expressions to their simplest form, making them easier to interpret and use.
When a function is expressed in a complicated form, simplifying it generally helps to clarify its behavior and calculations.

• Simplification makes it easier to solve equations and understand the function's domain and range.
• It often involves combining like terms and reducing coefficients.

The process is crucial in ensuring that the equations are workable, especially in tasks like graphing or solving higher-level mathematics problems. Here, in our exercise, the goal was to exemplify this by turning the expression (3 imes (x+1) - 2) into (3x + 1).

Distributive Property

The distributive property is a fundamental mathematical principle crucial for simplifying algebraic expressions. It especially applies when you need to eliminate parentheses. The property states that multiplying a number by a sum is the same as (and simpler than) multiplying each addend by the number and then adding the products.
For instance:

• The distributive property is expressed as: \[ a(b+c) = ab + ac \]
• In our case: \[ 3(x+1) = 3x + 3 \]

This logical move helps to remove parentheses, leading to expressions that are more straightforward and workable in further computation.
Mastering the distributive property is key for solving many types of algebraic problems, involving polynomial operations.

Polynomial Functions

Polynomial functions are an important category of mathematical functions.

• They are expressed in the form: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where the numbers \( a_n, a_{n-1}, ..., a_0 \) are constants.
• The coefficients can be any real numbers, and the power of \( x \) (the degree of the polynomial) determines its complexity.
• A linear polynomial, like the one in the exercise, appears as \( g(x) = 3x - 2 \).

Polynomial functions are foundational in algebra and calculus because of their simple structure and predictable behavior.
Understanding how to manipulate and simplify polynomials, like transforming \( g(x+1) = 3x + 1 \), is critical for anyone studying mathematics thoroughly. They form the basis of more complex algebraic structures and offer insight into mathematical modeling.