Problem 68 - College Algebra

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

f(-2) = 8, f(-1) = 6, f(0) = 4, f(1) = 2, f(2) = 0.

Step 1: Evaluate f(-2)

To find \(f(-2)\), substitute \(-2\) for \(x\) in the function \(f(x) = 4-2x\). This yields: \(f(-2) = 4 - 2(-2) = 4 + 4 = 8\).

Step 2: Evaluate f(-1)

Substitute \(-1\) for \(x\) in the function \(f(x) = 4-2x\). This yields: \(f(-1) = 4 - 2(-1) = 4 + 2 = 6\).

Step 3: Evaluate f(0)

Substitute \(0\) for \(x\) in the function \(f(x) = 4-2x\). This yields: \(f(0) = 4 - 2(0) = 4\).

Step 4: Evaluate f(1)

Substitute \(1\) for \(x\) in the function \(f(x) = 4-2x\). This yields: \(f(1) = 4 - 2(1) = 4 - 2 = 2\).

Step 5: Evaluate f(2)

Substitute \(2\) for \(x\) in the function \(f(x) = 4-2x\). This yields: \(f(2) = 4 - 2(2) = 4 - 4 = 0\).

Key Concepts

Linear Functions

Linear functions are essential in mathematics due to their simplicity and wide range of applications. They have a straightforward form:

• They can be written as: \( f(x) = mx + b \)
• Where \( m \) represents the slope and \( b \) is the y-intercept
• The function produces a straight line when graphed on a coordinate plane

In the given function \( f(x) = 4 - 2x \), the expression is linear because it follows the general form. Here, \( m = -2 \) and \( b = 4 \). This reflects a line with a negative slope indicating it will descend from left to right on the graph.
Understanding linearity helps in predicting the behavior of the line based on the slope and intercept values.

Substitution in Functions

Substitution is a technique in algebra that helps in evaluating mathematical expressions for specific values. When dealing with functions, substitution involves replacing a variable, usually \( x \), with a given number to calculate the function's value.

• Begin with the function equation
• Substitute the provided value into the function
• Simplify the equation to find the result

For example, in evaluating \( f(-2) \) from \( f(x) = 4 - 2x \), we substitute \(-2\) for \( x \), which means the equation changes to \( f(-2) = 4 - 2(-2) \). Simplifying results in \( 4 + 4 = 8 \).
Substitution helps us understand how the function behaves at different points, giving insights about changes in the entire function.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding them is crucial for simplifying and manipulating expressions in math. They can take different forms, and evaluating them often involves various algebraic operations such as addition, subtraction, multiplication, or division.

• A simple structure like \( f(x) = 4 - 2x \)
• Involves simple arithmetic operations on variables
• Helps in forming equations that model real-world scenarios

The expression \( 4 - 2x \) is straightforward, involving basic operations of subtraction and multiplication. To compute, substitute the values into the equation, then follow the order of operations.
Mastery in handling algebraic expressions lays the groundwork for more advanced topics in algebra and calculus.