Problem 41 - Algebra 2

Question

Find the value of \(f(x)=-3 x+2\) when \(x=2\)

Step-by-Step Solution

Answer

The value of \(f(2)\) is \(-4\).

Step 1: Substitute the Value of x into the Function

To find the value of the function for a specific value of \(x\), substitute \(x=2\) into the function \(f(x) = -3x + 2\).

Step 2: Calculate the Resulting Expression

After substitution, the function becomes \(f(2) = -3(2) + 2\). Now, perform the multiplication: \(-3 \times 2 = -6\).

Step 3: Perform Addition to Find the Function Value

Add the result from the multiplication to the constant in the function: \(-6 + 2 = -4\).

Key Concepts

Linear Functions

A linear function is a type of mathematical expression that describes a straight line when graphed on a coordinate plane. In general, a linear function can be written in the form of \(f(x) = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. The slope \(m\) indicates how steep the line is, and the y-intercept \(b\) tells us where the line crosses the y-axis.
Linear functions are important in many areas of mathematics and real-world applications due to their simplicity and predictability. They are also helpful in modeling situations where there is a constant rate of change.
In the linear function given in the problem, \(f(x) = -3x + 2\), the slope \(m\) is \(-3\) and the y-intercept \(b\) is \(2\). This means for every unit increase in \(x\), \(f(x)\) decreases by 3 units, and when \(x = 0\), the value of \(f(x)\) is 2.

Substitution

Substitution is a fundamental technique used in algebra to solve functions or equations. It involves replacing a variable with a given number in an expression or equation so we can simplify and solve it. Substitution is most commonly used when evaluating functions, like finding the value of a function at a specific point.
In the given exercise, the value of \(x\) to be substituted into the function \(f(x)= -3x + 2\) is \(2\). To substitute, we replace every occurrence of \(x\) in the equation with the number \(2\). This changes the expression to \(f(2) = -3(2) + 2\).
This substitution enables us to compute the value of the function at that specific point, helping us understand how the variable \(x\) affects the outcome of the function.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations which include addition, subtraction, multiplication, and division. These operations are the building blocks for more complex mathematical problems.
In the solution to the given problem, two main arithmetic operations are performed: multiplication and addition.

• **Multiplication:** The first step involves multiplying \(-3\) by \(2\), calculated as \(-3 \times 2 = -6\).
• **Addition:** Following multiplication, the result is added to \(2\), giving us \(-6 + 2 = -4\).

These arithmetic steps are essential to reaching the final solution. Understanding each operation helps in making precise calculations and is crucial when performing algebraic manipulations. Arithmetic operations form the basis upon which additional math skills are built.