Problem 48 - Intermediate Algebra

Question

Is \(f(x)=(3 x-2)-(2 x+1)\) a linear function? Explain your answer.

Step-by-Step Solution

Answer

Yes, it's a linear function.

Step 1: Identify the components of the function

The function given is in the form of a difference: \( f(x) = (3x - 2) - (2x + 1) \). Our task is to simplify this function and determine whether it's linear.

Step 2: Simplify the function

Start by distributing the subtraction across the second set of parentheses: \( f(x) = 3x - 2 - 2x - 1 \). This means you have to subtract both terms inside the second parentheses from the first set of terms.

Step 3: Combine like terms

Combine the \( x \) terms and the constants separately: \( 3x - 2x = x \) and \(-2 - 1 = -3 \). Thus, the simplified function is \( f(x) = x - 3 \).

Step 4: Determine if the function is linear

A linear function is one that can be written in the form \( f(x) = ax + b \). Our simplified function is \( f(x) = x - 3 \), which is clearly in the form \( 1x + (-3) \), with \( a = 1 \) and \( b = -3 \). This confirms that the function is linear.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential step in solving and understanding mathematical problems. In this exercise, we started with the function \( f(x) = (3x - 2) - (2x + 1) \). The goal was to simplify it to see if it's a linear function. Here's how we do it:

• Identify and distribute any operations, like subtraction or distribution across terms.
• Remove parentheses carefully, remembering that to subtract means to subtract each element in parentheses.
• In our case, subtract \((2x+1)\) by changing it to \(-2x-1\), allowing us to deal with individual terms.

The result is \( f(x) = 3x - 2 - 2x - 1 \). By tackling expressions piece-by-piece, you're reducing complexity and making it easier to spot solutions. This forms the basis for simplifying complex expressions into manageable forms.

Function Notation

Understanding function notation is crucial for interpreting mathematical functions and their behaviors. The notation \( f(x) \) represents a function where \( f \) is the symbol for the function, and \( x \) is the variable. This is equivalent to the input, which gives the output when substituted into the expression.
In our exercise, \( f(x) = (3x - 2) - (2x + 1) \) shows that different operations affect \( x \) independently within the function.

• Function notation simplifies expressing various relationships between inputs and outputs, helping us track how they interact.
• The concept of a function as a 'machine' translates well to coding and algorithmic thinking.

Function notation helps streamline communication in math, offering a clear way to express any transformations or operations applied to the variable.

Combining Like Terms

Combining like terms is an essential part of simplifying expressions, particularly in algebra. In our example, after removing the parentheses, we ended up with \( f(x) = 3x - 2 - 2x - 1 \). To simplify, we needed to combine like terms:

• The terms involving \( x \) (i.e., \( 3x \) and \( -2x \)) are like terms because they each contain the variable \( x \).
• After combining like terms: \( 3x - 2x = x \).
• Similarly, the constant terms \(-2\) and \(-1\) are combined: \(-2 - 1 = -3\).

The result is a simplified form: \( f(x) = x - 3 \). This alone illustrates the power of combining like terms, reducing redundancy and making equations easier to work with. It turns a potentially confusing expression into something straightforward and easy to manage.