Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=3 x-4 ; x=-2$$

Step-by-Step Solution

Answer

The value of the function at \(x = -2\) is \(-10\).

Step 1: Evaluate the function at specified x-value

To find the value of the function \(f(x) = 3x - 4\) at \(x = -2\), we substitute \(-2\) into the function. This gives us:\\[f(-2) = 3(-2) - 4\]

Step 2: Calculate the expression

We now solve the expression by performing the multiplication and subtraction.\Multiply \(3\) by \(-2\): \\[3(-2) = -6\]\Next, we subtract \(4\) from \(-6\): \\[-6 - 4 = -10\]

Step 3: Write down the final result

The function value at \(x = -2\) is calculated to be \(-10\). Therefore, \(f(-2) = -10\).

Key Concepts

Linear Function

A linear function is one of the simplest types of functions, characterized by a straight line when graphed on a coordinate plane. The general form of a linear function is given by:- \(f(x) = mx + b\)where:- \(m\) represents the slope of the line, determining its steepness,- \(b\) is the y-intercept, or the point where the line crosses the y-axis.In our original problem, the function \(f(x) = 3x - 4\) is linear. Here, \(m = 3\) indicates the line rises 3 units for every unit moved to the right, and \(b = -4\) suggests the line crosses the y-axis at -4. The beauty of linear functions is in their predictability and simplicity, making them a great starting point for understanding more complex functions later.Linear functions are commonly used in real-life situations such as calculating cost, predicting future trends, and understanding rates of change.Understanding the nature of linear functions allows us to quickly evaluate and interpret situations where relationships are straightforward and consistent.

Substitution

Substitution is an algebraic method used to solve for unknown variables by replacing them with known values. This method is extensively used, especially in evaluating functions.To perform substitution, you:

• Identify the variable to substitute.
• Replace this variable with the given value.
• Solve the resulting expression.

In our example, the task was to find \(f(-2)\) for the function \(f(x) = 3x - 4\). Here, substitution involves replacing \(x\) with \(-2\) in the equation:- \(f(-2) = 3(-2) - 4\)Using substitution allows us to determine specific values of functions efficiently. It's particularly useful for functions like linear functions that consistently change due to their constant slope and intercept terms. This technique is pivotal in making equations or problems easier to handle and providing concise solutions.

Arithmetic Operations

Arithmetic operations are basic mathematical operations that include addition, subtraction, multiplication, and division. They form the core of evaluating any algebraic expression after substitution has occurred.In our function evaluation, after substituting \(x = -2\) into the linear function \(f(x) = 3x - 4\), we performed arithmetic operations:

• Multiplication: Initially, \(3\) times \(-2\) gives \(-6\).
• Subtraction: Then, subtracting \(4\) from \(-6\) results in \(-10\).

These operations are straightforward yet crucial for simplifying expressions to reach the solution effectively. Each step relies on understanding and executing these operations correctly, showcasing the importance of arithmetic in processing and solving mathematical functions accurately. Mastery of arithmetic operations is fundamental for students to progress in algebra and more advanced areas of math.