Problem 35 - Algebra 2

Question

Find each value if \(f(x)=3 x-5\) and \(g(x)=x^{2}-x\) \(f(-3)\)

Step-by-Step Solution

Answer

\( f(-3) = -14 \)

Step 1: Identify the function

The problem gives us the function \( f(x) = 3x - 5 \). We need to use this function to find \( f(-3) \).

Step 2: Substitute \( x = -3 \) into the function

Replace \( x \) in the function \( f(x) \) with \( -3 \). This gives us: \[f(-3) = 3(-3) - 5\]

Step 3: Perform the multiplication

Calculate \(3 \times -3\). This step gives: \[3(-3) = -9\]

Step 4: Complete the subtraction

Substitute the result from the previous step into the function: \[f(-3) = -9 - 5\]Performing the subtraction gives: \[f(-3) = -14\]

Key Concepts

Understanding Linear Functions

A linear function is a simple yet fundamental concept in mathematics. It's an algebraic function that forms a straight line when graphed on a coordinate plane. The standard form of a linear function is expressed as \( f(x) = mx + b \), where:

• \( m \) represents the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In the context of our exercise, the function \( f(x) = 3x - 5 \) is linear. Here, the slope \( m = 3 \) signifies that for every unit increase in \( x \), \( f(x) \) increases by 3 units. Meanwhile, the y-intercept here is -5, meaning the line crosses the y-axis at -5. It's crucial to grasp these two components, as they describe the entire behavior of the line. Linear functions are key in solving real-world problems because they model relationships with a constant rate of change.

Exploring the Substitution Method

The substitution method is a straightforward technique used in function evaluation. It involves replacing a variable with a given number to solve an equation or evaluate a function. This method is particularly useful for finding specific outputs of a function.In the exercise, we employed the substitution method to determine \( f(-3) \) from the function \( f(x) = 3x - 5 \). By substituting \( -3 \) for \( x \), we transformed the original function into a specific equation: \( 3(-3) - 5 \). This approach is helpful because it systematically narrows down a general function to a specific answer.The substitution method applies beyond this example, being widely used in solving equations, especially in algebra, calculus, and other mathematical fields. It simplifies problem-solving by allowing students to isolate and solve for unknowns in manageable steps.

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical relationships. In the function \( f(x) = 3x - 5 \), \( 3x - 5 \) is an algebraic expression.It's crucial to understand each component of an algebraic expression:

• **Terms**: The parts of an expression separated by plus or minus signs. In our case, \( 3x \) and \( -5 \) are the terms.
• **Coefficients**: These are the numbers multiplied by the variables. Here, 3 is the coefficient of \( x \).
• **Constants**: Numbers on their own, like -5, are constants.

Analyzing algebraic expressions involves operations like addition, subtraction, multiplication, and sometimes division. In our exercise, multiplying 3 by -3 and subtracting 5 were necessary steps to evaluate \( f(-3) \). Understanding how to manipulate and evaluate expressions is essential in algebra as it lays the groundwork for solving more complex mathematical problems.