Problem 25 - Algebra 1: Concepts and Skills

Question

Evaluate the function when \(x=2, x=0\) and \(x=-2\) $$ f(x)=3 x $$

Step-by-Step Solution

Answer

The solutions are: \(f(2) = 6\), \(f(0) = 0\), \(f(-2) = -6\)

Step 1: Evaluate at \(x=2\)

Plug in the \(x\)-value into the function: \( f(2) = 3 \times 2 = 6\)

Step 2: Evaluate at \(x=0\)

Substitute \(x=0\) into the function: \( f(0) = 3 \times 0 = 0 \)

Step 3: Evaluate at \(x=-2\)

Replace \(x\) with \(-2\) in the function: \( f(-2) = 3 \times -2 = -6 \)

Key Concepts

Linear Functions

A linear function is a type of function defined by an equation that creates a straight line when graphed on a coordinate plane. One of the simplest forms of a linear function is given by the equation:

• \( f(x) = mx + b \)

Here, \( m \) represents the slope of the line, which determines how steep the line is, and \( b \) is the y-intercept, where the line crosses the y-axis.
In the given exercise, the function \( f(x) = 3x \) defines a linear function without a y-intercept (the \( b \) term is zero). Linear functions like this one, depicted solely by \( f(x) = mx \), pass through the origin \((0,0)\).
This means the rate at which the function increases or decreases is constant, and that rate is dictated by the coefficient \( m \), referred to as the slope. Here, the slope \( m \) is 3, indicating that for every unit increase in \( x \), \( f(x) \) increases by 3 units.

Substitution Method

The substitution method is a straightforward mathematical technique used to evaluate functions at specific points. It involves replacing the variable in a function with a given value. In the exercise, you're asked to substitute different values of \( x \) into the linear function \( f(x) = 3x \).
To understand it better:

• Firstly, identify the function: \( f(x) = 3x \).
• Then, substitute the given values of \( x \) into the function one at a time.

For example, if \( x = 2 \):

• Replace \( x \) with 2 in the function: \( f(2) = 3 \times 2 = 6 \).

The substitution method helps to simplify complex expressions and determine specific outputs of a function efficiently.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations like addition, subtraction, multiplication, and division, structured into meaningful mathematical sentences. In the context of the exercise, \( f(x) = 3x \) is an algebraic expression where

• \( x \) is a variable, representing any number.
• \( 3 \) is a constant that multiplies \( x \).

To evaluate an algebraic expression like \( 3x \), substitute the value of the variable with specific numbers, such as 2, 0, or -2.
Algebraic expressions are foundational in algebra because they facilitate finding relationships between quantities and can often be manipulated through arithmetic operations.