Problem 53 - A Graphical Approach to College Algebra

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 \( f(-2) \) is \(-10\).

Step 1: Understand the Function

The function given is a linear function defined as \( f(x) = 3x - 4 \). This means for any value of \( x \), the function \( f(x) \) is calculated by multiplying \( x \) by 3 and then subtracting 4.

Step 2: Substitute the Given Value

We need to find \( f(x) \) for \( x = -2 \). Substitute \( x = -2 \) into the function: \( f(-2) = 3(-2) - 4 \).

Step 3: Calculate the Expression

First, perform the multiplication: \( 3 \times (-2) = -6 \). This results in \( f(-2) = -6 - 4 \).

Step 4: Simplify the Result

Complete the calculation: \( -6 - 4 = -10 \). Thus, \( f(-2) = -10 \).

Key Concepts

Linear Function

A linear function is a type of function where the relationship between the input, typically denoted by \(x\), and the output, \(f(x)\), is linear, meaning it forms a straight line when graphed. The general formula for a linear function is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This structure is simple and predictable:

• Slope \(m\): Indicates how steep the line is. It tells you how much \(f(x)\) changes for each unit increase in \(x\).
• Y-intercept \(b\): The point where the line crosses the y-axis, which is the value of \(f(x)\) when \(x = 0\).

In our exercise, the function \(f(x) = 3x - 4\) has a slope of 3 and a y-intercept of -4, showing that for each increase of 1 in \(x\), \(f(x)\) increases by 3. This predictable pattern is what defines the "linear" in linear functions.

Substitution Method

The substitution method involves replacing a variable with a given value to simplify and evaluate functions or equations. This is a fundamental technique in algebra that simplifies calculations and helps find specific outputs for certain inputs.

• When you know the value for \(x\), you substitute it directly into the function.
• This allows you to calculate a concrete result for \(f(x)\).

In our example, we substitute \(x = -2\) into the function \(f(x) = 3x - 4\) to find \(f(-2)\). Once substituted, the expression becomes \(3(-2) - 4\), and you continue with arithmetic to find the result. Substitution makes it possible to pinpoint exact values and determine the behavior of mathematical functions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They don't usually include an "equals" sign unless they're part of an equation.

• Variables: Symbols used to represent unknown values, often denoted as \(x, y, z\), and so forth.
• Operations: Include addition, subtraction, multiplication, and division, which are used to construct expressions.

In the given linear function \(f(x) = 3x - 4\), the expression consists of the term \(3x\) (where 3 is a coefficient and \(x\) is a variable) and the constant -4. These are combined using basic arithmetic operations to form the expression. Evaluating algebraic expressions like these is key to solving real-world problems and understanding mathematical relationships.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. This involves substituting values into the function and performing necessary calculations.

• Identify the function structure, such as \(f(x) = mx + b\).
• Insert the given input value into the function.
• Simplify using basic arithmetic to find the result.

For example, to evaluate \(f(x) = 3x - 4\) at \(x = -2\), substitute \(-2\) for \(x\), which gives \(f(-2) = 3(-2) - 4 = -10\). Function evaluation is essential for understanding how a function operates at specific points, making it a critical skill for solving mathematical problems effectively.