Evaluating And Solving Linear Functions A Comprehensive Guide To G(x) = -3.3x - 4
This article delves into the intricacies of the linear function g(x) = -3.3x - 4. We will explore how to evaluate this function for specific input values and how to determine the input value that produces a desired output. This comprehensive analysis will provide a solid understanding of linear functions and their practical applications.
(a) Evaluating g(15): A Step-by-Step Guide
In this section, we focus on evaluating the function g(x) = -3.3x - 4 when x = 15. This process involves substituting the value of x into the function and performing the necessary arithmetic operations. Understanding how to evaluate functions is a fundamental skill in mathematics, particularly in algebra and calculus. It allows us to determine the output of a function for any given input, which is crucial for understanding the function's behavior and its applications in various fields.
To find g(15), we begin by replacing x with 15 in the function's equation:
g(15) = -3.3(15) - 4
Next, we perform the multiplication:
g(15) = -49.5 - 4
Finally, we complete the subtraction:
g(15) = -53.5
Therefore, the value of the function g(x) when x = 15 is -53.5. This means that the point (15, -53.5) lies on the graph of the function g(x). Evaluating functions at specific points is essential for graphing the function, identifying its intercepts, and understanding its slope and direction. Furthermore, this skill is crucial in solving equations and inequalities involving functions.
Evaluating functions also has numerous real-world applications. For example, if g(x) represents the cost of producing x units of a product, then g(15) would represent the cost of producing 15 units. Similarly, if g(x) represents the distance traveled by a car after x hours, then g(15) would represent the distance traveled after 15 hours. Understanding how to evaluate functions allows us to model and analyze various real-world phenomena.
The process of evaluating g(15) highlights the importance of order of operations in mathematics. We first performed the multiplication before the subtraction, following the PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) rule. This rule ensures that mathematical expressions are evaluated consistently, leading to accurate results. Mastering the order of operations is essential for success in mathematics and related fields.
In summary, evaluating g(15) involved substituting x = 15 into the function g(x) = -3.3x - 4 and performing the arithmetic operations. The result, g(15) = -53.5, represents the output of the function for the given input. This process demonstrates the fundamental concept of function evaluation and its importance in understanding and applying mathematical functions.
(b) Calculating g(-20): A Detailed Explanation
In this section, we will determine the value of the function g(x) = -3.3x - 4 when x = -20. This process is similar to the previous section, but it involves working with a negative input value. Understanding how to evaluate functions with negative inputs is crucial because it expands the domain of the function and allows us to model scenarios where negative values are meaningful. For example, in finance, negative values can represent debt or losses. In physics, negative values can represent direction or displacement.
To calculate g(-20), we substitute x with -20 in the function's equation:
g(-20) = -3.3(-20) - 4
Next, we perform the multiplication. Remember that multiplying two negative numbers results in a positive number:
g(-20) = 66 - 4
Finally, we complete the subtraction:
g(-20) = 62
Therefore, the value of the function g(x) when x = -20 is 62. This indicates that the point (-20, 62) lies on the graph of the function g(x). Evaluating functions with negative inputs allows us to understand the function's behavior in different regions of the coordinate plane and to model situations involving negative quantities.
Working with negative numbers requires careful attention to signs and order of operations. The multiplication of -3.3 and -20 resulted in a positive value, 66, which was then subtracted by 4. A common mistake is to incorrectly apply the negative sign, which can lead to an incorrect result. Practicing with various examples involving negative numbers can help solidify understanding and prevent errors.
The result g(-20) = 62 has a graphical interpretation as well. It means that when x is -20, the y-value on the graph of the function g(x) is 62. This point, (-20, 62), can be plotted on a coordinate plane, and it helps to visualize the overall shape and behavior of the function. Understanding the connection between the algebraic representation of a function and its graphical representation is a key aspect of mathematical literacy.
Furthermore, evaluating g(-20) can have practical interpretations depending on the context. If g(x) represents the temperature in degrees Celsius at time x hours before a certain event, then g(-20) would represent the temperature 20 hours before the event. If g(x) represents the profit of a company x months ago, then g(-20) would represent the profit 20 months ago. The ability to interpret the results of function evaluation in real-world contexts is a valuable skill.
In conclusion, calculating g(-20) involved substituting x = -20 into the function g(x) = -3.3x - 4 and performing the arithmetic operations. The result, g(-20) = 62, demonstrates the process of evaluating functions with negative inputs and highlights the importance of careful attention to signs and order of operations. This skill is essential for understanding the behavior of functions and their applications in various fields.
(c) Finding x when g(x) = -32.6: Solving for the Input
This section focuses on a different aspect of working with functions: finding the input value x that produces a specific output value. In this case, we want to find the value of x for which g(x) = -32.6. This involves solving an equation, which is a fundamental skill in algebra. Solving equations allows us to determine the input values that satisfy a given condition, which is crucial in many mathematical and real-world applications. For example, we might want to find the time it takes for a projectile to reach a certain height or the number of units that need to be sold to achieve a specific profit target.
To find the value of x such that g(x) = -32.6, we set the function g(x) = -3.3x - 4 equal to -32.6 and solve for x:
-3.3x - 4 = -32.6
First, we add 4 to both sides of the equation to isolate the term with x:
-3.3x = -28.6
Next, we divide both sides of the equation by -3.3 to solve for x:
x = -28.6 / -3.3
Performing the division, we get:
x ≈ 8.6667
The problem asks us to round the answer to one decimal place. Therefore:
x ≈ 8.7
Thus, the value of x that makes g(x) = -32.6 is approximately 8.7. This means that the point (8.7, -32.6) lies on the graph of the function g(x). Finding the input value for a given output is the inverse process of evaluating a function, and it is a crucial skill in solving problems involving functions.
The process of solving for x involved using algebraic manipulations to isolate the variable. We used the properties of equality to add and divide both sides of the equation by the same value, ensuring that the equation remained balanced. Understanding these properties is essential for solving any algebraic equation.
The solution x ≈ 8.7 can be verified by substituting this value back into the original function g(x) and checking if the result is approximately -32.6:
g(8.7) = -3.3(8.7) - 4
g(8.7) = -28.71 - 4
g(8.7) = -32.71
This value is very close to -32.6, which confirms that our solution is accurate, considering the rounding we performed. This verification step is a good practice to ensure the correctness of the solution.
In conclusion, finding x such that g(x) = -32.6 involved solving an equation using algebraic manipulations. The result, x ≈ 8.7, represents the input value that produces the desired output. This process demonstrates the importance of solving equations in working with functions and highlights the connection between algebra and functions.
Conclusion: Mastering Linear Functions
In this article, we have explored the linear function g(x) = -3.3x - 4 in detail. We learned how to evaluate the function for specific input values, including positive and negative numbers. We also learned how to find the input value that produces a desired output by solving an equation. These skills are fundamental to understanding and applying linear functions in various contexts. Mastering linear functions is a crucial step in building a strong foundation in mathematics and its applications in science, engineering, economics, and many other fields. The ability to work with functions, evaluate them, and solve equations involving them is essential for problem-solving and critical thinking. Understanding the behavior of linear functions, including their slope, intercepts, and graphical representation, allows us to model and analyze real-world phenomena effectively. By practicing these skills and exploring different examples, you can develop a deep understanding of linear functions and their applications.
