Problem 68 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=4-2 x\)

Step-by-Step Solution

Answer

Function values are 8, 6, 4, 2, and 0 for \(x = -2, -1, 0, 1, 2\) respectively.

Step 1: Evaluate f(-2)

Substitute \(x = -2\) into the function \(f(x) = 4 - 2x\). So, \(f(-2) = 4 - 2(-2) = 4 + 4 = 8\).

Step 2: Evaluate f(-1)

Substitute \(x = -1\) into the function \(f(x) = 4 - 2x\). So, \(f(-1) = 4 - 2(-1) = 4 + 2 = 6\).

Step 3: Evaluate f(0)

Substitute \(x = 0\) into the function \(f(x) = 4 - 2x\). So, \(f(0) = 4 - 2(0) = 4\).

Step 4: Evaluate f(1)

Substitute \(x = 1\) into the function \(f(x) = 4 - 2x\). So, \(f(1) = 4 - 2(1) = 4 - 2 = 2\).

Step 5: Evaluate f(2)

Substitute \(x = 2\) into the function \(f(x) = 4 - 2x\). So, \(f(2) = 4 - 2(2) = 4 - 4 = 0\).

Key Concepts

Linear Functions

Linear functions are a fundamental concept in mathematics, representing a relationship where each input, or x-value, results in a corresponding output. In essence, they create a straight line when graphed. The generic form of a linear function is denoted as \( f(x) = mx + b \), where \( m \) denotes the slope, explaining how steep the line is, and \( b \) represents the y-intercept, indicating where the line passes through the y-axis. For instance, the function given in our original exercise, \( f(x) = 4 - 2x \), is a linear function. Here, the slope \( m \) is -2, meaning the line descends as the x-value increases. And the y-intercept is 4, so the line crosses the y-axis at \( y = 4 \). This function tells us how y decreases by 2 units for every one-unit increase in x. Understanding linear functions helps us identify constant rates of change. They are easy to work with due to this predictability and are widely used in many areas, from economics to physics.

Substitution Method

The substitution method is a handy tool used to find the value of a function at specific points. It involves replacing the variable (often \( x \)) in the function with the requested values. Essentially, it allows us to 'solve' the function for given values of x. Let's take a closer look at how it works:

• Start by identifying the function. In our example, that's \( f(x) = 4 - 2x \).
• Choose the specific x-values you need, such as -2, -1, 0, 1, and 2 in the given exercise.
• Replace the \( x \) in the equation with each value, one at a time.
• Perform the required calculations to solve for \( f(x) \).

For the function \( f(x) = 4 - 2x \), using the substitution method provided straightforward results: \( f(-2) = 8 \), \( f(-1) = 6 \), \( f(0) = 4 \), \( f(1) = 2 \), and \( f(2) = 0 \). This method helps us efficiently determine the outcome of a function at specific points.

Algebraic Operations

Algebraic operations are fundamental steps in solving equations, particularly in the context of functions. They include addition, subtraction, multiplication, and division. When evaluating functions like \( f(x) = 4 - 2x \), these operations play a crucial role. Here's how they apply:

• Substitution introduces values into an equation, requiring basic operations to simplify expressions. For example, when \( x = -2 \), substitution changes the function to \( 4 - 2(-2) \).
• Use multiplication as shown, where \(-2 \times -2 = 4\). Inclusion of negative numbers often flips the operation to addition.
• Addition within expressions involves directly summing results, such as \(4 + 4 = 8\) in \( f(-2) \).
• If necessary, subtraction is employed to reduce terms in the function, for example in \( f(1) = 4 - 2 \times 1 = 2 \).

Algebraic operations provide the foundation to interpret and solve linear functions effectively. They enable you to manipulate and simplify the expressions generated during function evaluation to arrive at precise solutions.