Problem 42 - Algebra 2

Question

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(f(-1)\)

Step-by-Step Solution

Answer

\( f(-1) = -4 \).

Step 1: Understand the Function Definition

First, identify the function you need to evaluate. In this case, you are given the function \( f(x) = 6x + 2 \).

Step 2: Substitute the Input Value into the Function

Substitute \( x = -1 \) into the function \( f(x) = 6x + 2 \). This yields \( f(-1) = 6(-1) + 2 \).

Step 3: Simplify the Expression

Calculate the expression. Start by multiplying: \( 6(-1) = -6 \). Then, add 2 to this result: \( -6 + 2 = -4 \).

Step 4: Write the Final Result

The calculated value of the function when \( x = -1 \) is \( f(-1) = -4 \).

Key Concepts

Function Notation

Function notation is a way to represent functions in a clear and concise manner. It typically uses the letter 'f' and symbolizes the relationship between the input, 'x', and the output, 'f(x)'. In the given exercise, we look at two functions:

• The first function, noted as \( f(x) = 6x + 2 \), defines \( f \) such that the output is dependent on the input \( x \).
• The second function is \( g(x) = 3x^2 - x \).

Function notation helps us quickly understand what operation needs to be performed on 'x' to get the desired output. It also makes it easy to reference specific function values, such as \( f(-1) \), which instructs us to evaluate the function at the input of \( -1 \). This notation is instrumental in all areas of calculus and algebra as it provides clarity and precision in describing mathematical relationships.

Substitution

Substitution is a fundamental process used in mathematics to solve equations and evaluate functions by replacing variables with specific values. In the context of our exercise, substitution involves the following steps:

• Identify the value to substitute: In this exercise, we need to find \( f(-1) \).
• Replace the variable \( x \) in the function \( f(x) = 6x + 2 \) with the number \( -1 \). This is shown as \( f(-1) = 6(-1) + 2 \).

Substitution is crucial because it allows us to determine the exact value of a function for a given input. This process is not just limited to plugging in numbers; it can also be used to substitute other expressions or variables, making it a versatile tool in problem-solving. The precision of substitution ensures that each step in the evaluation is clear and correct, helping to avoid mistakes.

Linear Functions

Linear functions are mathematical expressions that create straight lines when plotted on a graph. They take the general form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Let's break it down using our function \( f(x) = 6x + 2 \):

• The slope \( m = 6 \) tells us how steep the line is. For every increase of 1 in \( x \), \( f(x) \) increases by 6.
• The y-intercept \( b = 2 \) indicates the point where the line crosses the y-axis.

Linear functions are simple yet powerful tools in mathematics. They have constant rates of change and can be used to model a variety of real-world situations, such as predicting costs or calculating distances. Understanding linear functions helps in interpreting data trends and making informed decisions.

Simplification

Simplification involves reducing complexity to arrive at a more manageable and comprehensible form. In mathematical operations, we simplify expressions to find the most straightforward solution. During our function evaluation, we simplified as follows:

• After substituting \( x = -1 \) into \( f(x) = 6x + 2 \), we had \( 6(-1) + 2 \).
• Multiplying gives \( -6 \), and then adding 2 results in \( -4 \).

Simplifying ensures that our final answer, \( f(-1) = -4 \), is accurate and clear. By breaking down each operation step-by-step, simplification aids in understanding how complex expressions or equations can be resolved systematically. This practice is foundational in both solving equations and verifying solutions in algebra and calculus.