Evaluating Linear Functions G(x) = 5x + 14 A Comprehensive Guide

by THE IDEN 65 views

In the realm of mathematics, linear functions hold a prominent position as fundamental building blocks. These functions, characterized by their straight-line graphs, serve as powerful tools for modeling real-world phenomena and making predictions. Among the various ways to represent linear functions, the equation g(x) = 5x + 14 stands out as a clear and concise expression. This particular function embodies the essence of linearity, where the output g(x) varies directly with the input x, with a constant rate of change represented by the coefficient 5. The constant term 14, known as the y-intercept, signifies the point where the line intersects the vertical axis.

The ability to evaluate linear functions is crucial for understanding their behavior and applying them effectively. Evaluating a function involves substituting a specific value for the input variable x and then performing the necessary arithmetic operations to determine the corresponding output g(x). This process allows us to pinpoint specific points on the line represented by the function, providing valuable insights into its overall trajectory. In this comprehensive guide, we will delve into the evaluation of the linear function g(x) = 5x + 14, exploring practical examples and unraveling the underlying concepts.

Evaluating g(x) at x = 2

To begin our exploration, let's consider the task of evaluating the function g(x) = 5x + 14 at the specific input value of x = 2. This means we want to find the output value g(2), which represents the point on the line where the horizontal coordinate is 2. To accomplish this, we simply substitute x = 2 into the equation and perform the arithmetic operations.

Step 1: Substitution

We replace the variable x in the equation with the value 2:

g(2) = 5(2) + 14

Step 2: Multiplication

Next, we perform the multiplication operation:

g(2) = 10 + 14

Step 3: Addition

Finally, we add the two terms together:

g(2) = 24

Therefore, we have determined that g(2) = 24. This means that when the input is x = 2, the corresponding output of the function is g(2) = 24. Graphically, this represents the point (2, 24) on the line represented by the function g(x) = 5x + 14.

Visualizing the Evaluation

Imagine a coordinate plane with the x-axis representing the input values and the y-axis representing the output values. The linear function g(x) = 5x + 14 is depicted as a straight line that slopes upwards from left to right. When we evaluate the function at x = 2, we are essentially locating the point on this line where the x-coordinate is 2. Tracing a vertical line from x = 2 on the x-axis until it intersects the graph of the function, we find the corresponding y-coordinate to be 24. This visually confirms our calculation that g(2) = 24.

Evaluating g(x) at x = -11

Now, let's consider another example and evaluate the function g(x) = 5x + 14 at the input value of x = -11. This time, we want to find the output value g(-11), which corresponds to the point on the line where the horizontal coordinate is -11. The process remains the same: we substitute x = -11 into the equation and perform the arithmetic operations.

Step 1: Substitution

We replace the variable x with the value -11:

g(-11) = 5(-11) + 14

Step 2: Multiplication

Next, we perform the multiplication operation, keeping in mind the rules of signed numbers:

g(-11) = -55 + 14

Step 3: Addition

Finally, we add the two terms together, noting that we are adding a negative number to a positive number:

g(-11) = -41

Therefore, we have found that g(-11) = -41. This means that when the input is x = -11, the corresponding output of the function is g(-11) = -41. On the graph of the function, this represents the point (-11, -41).

Interpreting Negative Outputs

The fact that g(-11) = -41 indicates that this point lies below the x-axis on the coordinate plane. The negative output value signifies that the line represented by the function g(x) = 5x + 14 has a portion that extends into the third quadrant, where both x and y coordinates are negative. This is a characteristic feature of linear functions with a positive slope, as they tend to decrease as we move towards more negative x-values.

The Significance of Slope and Intercept

The linear function g(x) = 5x + 14 provides a clear illustration of the fundamental concepts of slope and y-intercept. The coefficient 5, which multiplies the input variable x, represents the slope of the line. The slope indicates the rate at which the output g(x) changes for every unit change in the input x. In this case, the slope of 5 signifies that for every increase of 1 in the value of x, the value of g(x) increases by 5.

The constant term 14, on the other hand, represents the y-intercept. The y-intercept is the point where the line intersects the y-axis, which occurs when x = 0. In the function g(x) = 5x + 14, the y-intercept is (0, 14), meaning that the line crosses the y-axis at the point where g(x) = 14.

Real-World Applications of Linear Functions

Linear functions are not merely abstract mathematical concepts; they have a wide range of applications in various real-world scenarios. They can be used to model relationships between different quantities that exhibit a constant rate of change. Here are a few examples:

  • Distance and Time: The distance traveled by an object moving at a constant speed can be modeled using a linear function. The speed represents the slope, and the initial distance represents the y-intercept.
  • Cost and Quantity: The total cost of purchasing a certain number of items at a fixed price per item can be represented by a linear function. The price per item is the slope, and any fixed costs are included in the y-intercept.
  • Temperature Conversion: The relationship between Celsius and Fahrenheit temperatures is linear. The conversion formula can be expressed as a linear function, allowing us to convert between the two scales.
  • Simple Interest: The amount of interest earned on a principal amount invested at a simple interest rate over time can be modeled using a linear function. The interest rate represents the slope, and the initial principal amount is the y-intercept.

The Power of Generalization

The process of evaluating the linear function g(x) = 5x + 14 at specific input values highlights a fundamental principle of mathematics: generalization. By understanding how to evaluate the function at particular points, we gain insight into its overall behavior and can make predictions about its output for any input value. This ability to generalize mathematical concepts is essential for solving problems and making informed decisions in various fields.

In this comprehensive guide, we have delved into the evaluation of the linear function g(x) = 5x + 14, exploring the concepts of slope, y-intercept, and their real-world applications. By evaluating the function at specific input values, such as x = 2 and x = -11, we gained a deeper understanding of its behavior and how it can be used to model linear relationships. The ability to evaluate functions is a cornerstone of mathematical proficiency, empowering us to solve problems, make predictions, and unravel the intricacies of the world around us. Linear functions, with their simplicity and versatility, serve as a powerful tool for mathematical modeling and analysis, paving the way for further exploration in the vast landscape of mathematics.