Problem 38 - Intermediate Algebra

Question

If \(f(x)=-3 x-4\), find \(f(-2), f(-1), f(3)\), and \(f(5)\).

Step-by-Step Solution

Answer

\( f(-2) = 2 \), \( f(-1) = -1 \), \( f(3) = -13 \), and \( f(5) = -19 \).

Step 1: Evaluate f(-2)

To find \( f(-2) \), substitute \( x = -2 \) into the function \( f(x) = -3x - 4 \). This gives us: \[ f(-2) = -3(-2) - 4 \] Simplifying inside the parentheses: \[ f(-2) = 6 - 4 \] Finally, subtract: \[ f(-2) = 2 \] So, \( f(-2) = 2 \).

Step 2: Evaluate f(-1)

To find \( f(-1) \), substitute \( x = -1 \) into the function \( f(x) = -3x - 4 \). This gives us: \[ f(-1) = -3(-1) - 4 \] Simplifying inside the parentheses: \[ f(-1) = 3 - 4 \] Finally, subtract: \[ f(-1) = -1 \] So, \( f(-1) = -1 \).

Step 3: Evaluate f(3)

To find \( f(3) \), substitute \( x = 3 \) into the function \( f(x) = -3x - 4 \). This gives us: \[ f(3) = -3(3) - 4 \] Simplifying inside the parentheses: \[ f(3) = -9 - 4 \] Finally, add: \[ f(3) = -13 \] So, \( f(3) = -13 \).

Step 4: Evaluate f(5)

To find \( f(5) \), substitute \( x = 5 \) into the function \( f(x) = -3x - 4 \). This gives us: \[ f(5) = -3(5) - 4 \] Simplifying inside the parentheses: \[ f(5) = -15 - 4 \] Finally, add: \[ f(5) = -19 \] So, \( f(5) = -19 \).

Key Concepts

Understanding Linear Functions

Linear functions are one of the foundational concepts in algebra and mathematics in general. They are called "linear" because their graph is a straight line. A linear function typically has the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants.

• "\( m \)" represents the slope of the line. It tells us how steep the line is and in which direction it moves.
• "\( b \)" is the y-intercept. This is the point on the graph where the line crosses the y-axis.

In our example of the function \( f(x) = -3x - 4 \), the slope \( m \) is -3, indicating that the line decreases as the x-value increases. The y-intercept \( b \) is -4, which tells us that the line crosses the y-axis at -4.
Linear functions are straightforward to evaluate. You simply substitute the x-value into the function and solve. This process will always yield a unique y-value, making linear functions predictable and simple to work with.

Introduction to the Substitution Method

The substitution method is a straightforward technique used to evaluate functions, solve equations, and more. It involves replacing a variable with a given value or expression to simplify a problem. The substitution method often comes into play when we need to evaluate a function at a specific point.
To use substitution with a function like \( f(x) = -3x - 4 \), follow these easy steps:

• Identify the x-value you need to evaluate, such as -2, -1, 3, or 5.
• Substitute this x-value into the function, replacing every instance of \( x \) with the number.
• Simplify the expression to find \( f(x) \).

For example, to find \( f(-2) \), you substitute \(-2\) into \( f(x) = -3x - 4 \), leading to \( f(-2) = -3(-2) - 4 = 6 - 4 = 2 \).
This method is particularly useful in algebra for finding specific function values and solving real-world problems quickly and accurately.

Exploring Intermediate Algebra

Intermediate algebra builds upon basic algebra concepts and prepares students for more advanced topics in mathematics. It covers various topics such as functions, equations, and inequalities. Understanding these concepts is pivotal for success in subsequent math courses.
Within intermediate algebra, linear functions like \( f(x) = -3x - 4 \) are essential. Mastery of how to evaluate and graph these functions can help students solve real-life problems, understand mathematical relationships better, and lay a solid foundation for calculus.

• It provides tools to solve multi-step equations and inequalities.
• Offers strategies for understanding and manipulating algebraic expressions.
• Includes learning about function properties, which is fundamental for analyzing more complicated functions later.

Practicing these skills helps in developing quantitative reasoning, which is valuable not only in mathematics but also in various scientific and engineering disciplines. This branch of algebra serves as a bridge to more advanced studies and practical applications in various fields.