Problem 77 - Algebra 2

Question

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ g(x)+g(x) $$

Step-by-Step Solution

Answer

The simplified form of \(g(x) + g(x)\) is \(x + 14\).

Step 1: Identifying the function

The function that is being simplified is \(g(x) + g(x)\), or equivalently, \(2 \cdot g(x)\). The function \(g(x)\) is given by \(g(x) = \frac{1}{2} x + 7\).

Step 2: Multiplication of function

We need to multiply the entire function \(g(x)\) by 2. Doing so, we get \(2 \cdot g(x) = 2 \cdot ( \frac{1}{2} x + 7 )\).

Step 3: Simplification of function

Distribute the 2 to each term inside the parentheses, we get \(2 \cdot \frac{1}{2} x + 2 \cdot 7\), which simplifies to \(x + 14\)

Key Concepts

Functions

In mathematics, functions are essential building blocks that establish relationships between two quantities, typically between variables like \( x \) and some output. A function, denoted by \( f(x) \), often involves mathematical operations applied to \( x \) to yield a result. For example, in our exercise, three different functions are given: \( f(x) = 4x \), \( g(x) = \frac{1}{2}x + 7 \), and \( h(x) = |-2x + 4| \). Each one of these functions demonstrates a specific way of manipulating the input \( x \) to get an output.Functions can be combined, added, or even multiplied. The task in the original exercise involves simplifying the expression \( g(x) + g(x) \). This operation can be perceived as either adding two identical functions or multiplying a function by 2. Understanding how to combine functions is crucial in algebra and allows you to see how simple or complex relationships develop between variables. Functions are fundamental, and mastering them opens doors to understanding other mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value or set of values. They are the language of algebra, allowing us to communicate mathematical ideas clearly. In our example, \( g(x) = \frac{1}{2}x + 7 \) is an algebraic expression that defines a function.An algebraic expression can include:

• Variables like \( x \).
• Coefficients like \( \frac{1}{2} \) in \( \frac{1}{2}x \).
• Constants like 7.
• Operations like addition, subtraction, multiplication, and division.

To simplify algebraic expressions, especially when functions are involved, it's important to combine like terms and apply mathematical rules. During simplification, expressions become less complicated but equivalent in value. This makes them easier to work with and understand for further operations.

Distributive Property

The distributive property is a fundamental algebraic property used to simplify expressions and solve equations. It states that a factor multiplying a sum of terms can be distributed to each addend, facilitating simplification. In mathematical terms, it is written as \( a(b + c) = ab + ac \).In our example with the function \( g(x) = \frac{1}{2}x + 7 \), when we simplify the expression \( g(x) + g(x) \), or \( 2 \cdot g(x) \), we apply the distributive property. Here's how:

• We begin with \( 2 \cdot (\frac{1}{2}x + 7) \).
• By applying the distributive property, multiply each term inside the parenthesis by 2, resulting in \( 2 \cdot \frac{1}{2}x + 2 \cdot 7 \).
• This simplifies to \( x + 14 \), as \( 2 \cdot \frac{1}{2}x \) equals \( x \) and \( 2 \cdot 7 \) becomes \( 14 \).

Utilizing the distributive property allows us to break down complex expressions into more manageable parts, making algebra problems less daunting and easier to solve.