Problem 69 - Algebra 1

Question

Evaluate the function for \(x=0,1,2,3,\) and 4. $$f(x)=-x+9 \quad$$

Step-by-Step Solution

Answer

The function \(f(x) = -x + 9\) evaluates to 9, 8, 7, 6, and 5 for \(x = 0, 1, 2, 3, 4\) respectively.

Step 1: Initialization

The function is given by \(f(x) = -x + 9\). We will evaluate this function for \(x = 0, 1, 2, 3, 4\).

Step 2: Evaluate \(f(x)\) at \(x=0\)

Replace \(x\) in \(f(x) = -x + 9\) with 0 to find \(f(0) = -(0) + 9 = 9\). So, when \(x = 0\), \(f(x) = 9\).

Step 3: Evaluate \(f(x)\) at \(x=1\)

Replace \(x\) in \(f(x) = -x + 9\) with 1 to find \(f(1) = -(1) + 9 = 8\). So, when \(x = 1\), \(f(x) = 8\).

Step 4: Evaluate \(f(x)\) at \(x=2\)

Replace \(x\) in \(f(x) = -x + 9\) with 2 to find \(f(2) = -(2) + 9 = 7\). So, when \(x = 2\), \(f(x) = 7\).

Step 5: Evaluate \(f(x)\) at \(x=3\)

Replace \(x\) in \(f(x) = -x + 9\) with 3 to find \(f(3) = -(3) + 9 = 6\). So, when \(x = 3\), \(f(x) = 6\).

Step 6: Evaluate \(f(x)\) at \(x=4\)

Replace \(x\) in \(f(x) = -x + 9\) with 4 to find \(f(4) = -(4) + 9 = 5\). So, when \(x = 4\), \(f(x) = 5\).

Key Concepts

Linear Functions

A linear function is a type of algebraic expression where the highest power of the variable is one. In simple terms, it forms a straight line when graphed on a coordinate plane. Linear functions can be written in the form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• The slope \( m \) determines the steepness or tilt of the line. For example, in the function \( f(x) = -x + 9 \), the slope is \(-1\), indicating the line decreases at a constant rate.
• The y-intercept is the point where the line crosses the y-axis. Here, it is \(9\), meaning the line crosses the y-axis at the point \((0, 9)\).

Understanding linear functions is key in algebra as they appear frequently in various mathematical problems and models.

Evaluation Steps

To evaluate a function, you substitute specific values for the variable \( x \) into the given algebraic expression and then compute the result. Let's look at evaluating the function \( f(x) = -x + 9 \):

• Start by replacing \( x \) with the first value. For example, when \( x = 0 \), you'll have \( f(0) = -0 + 9 = 9 \).
• Continue this process by inserting each subsequent value for \( x \) into the function. When \( x = 1 \), you evaluate it to get \( f(1) = -1 + 9 = 8 \).
• Repeat these steps for \( x = 2, 3, \text{ and } 4 \), which yields \( f(2) = 7 \), \( f(3) = 6 \), \( f(4) = 5 \).

By following each step systematically, you ensure accurate results for function evaluations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like \( x \)), and arithmetic operations such as addition, subtraction, multiplication, and division.

• In \( f(x) = -x + 9 \), the expression consists of the variable \( x \) with a coefficient of \(-1\), and a constant term of \( 9 \).
• To simplify or solve algebraic expressions, you apply arithmetic rules and substitute variables with numbers as needed in evaluation processes.
• These expressions can represent real-world situations, such as calculating distances, costs, or other measurements by inputting the necessary data.

Mastery of algebraic expressions lays the foundation for more advanced algebra concepts and problem-solving.