Analyzing The Function G(x) = X³ + 6x² + 12x + 8 And Determining Its Values

In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. This article delves into the intricacies of a specific function, g(x) = x³ + 6x² + 12x + 8, and explores its behavior through a combination of algebraic analysis and numerical evaluation. We will leverage the provided table of values to gain insights into the function's characteristics and learn how to determine the function's value for any given input.

Understanding the Function g(x) = x³ + 6x² + 12x + 8

The first step in understanding any function is to analyze its structure. Our function, g(x) = x³ + 6x² + 12x + 8, is a polynomial function of degree 3, also known as a cubic function. The general form of a cubic function is ax³ + bx² + cx + d, where a, b, c, and d are constants. In our case, a = 1, b = 6, c = 12, and d = 8. Polynomial functions are well-behaved, meaning they are continuous and differentiable everywhere. This allows us to use a variety of algebraic techniques to analyze their behavior. One crucial observation about g(x) is that it can be rewritten in a more compact form. Recognizing the coefficients, we can see that it is a perfect cube:

g(x) = x³ + 6x² + 12x + 8 = (x + 2)³

This rewritten form provides immediate insights into the function's roots and overall behavior. The root of the function is the value of x for which g(x) = 0. Setting (x + 2)³ = 0, we find that x = -2 is the only real root of the function, and it has a multiplicity of 3. This means the function touches the x-axis at x = -2 but does not cross it. The cubic nature of the function, along with the single real root, indicates that the function will increase monotonically as x increases. The rewritten form also simplifies the process of evaluating the function for different values of x. Instead of substituting into the cubic expression, we can simply add 2 to x and cube the result. This will be particularly useful when we use the provided table to understand the function's behavior.

Analyzing the Table of Values

The provided table offers a snapshot of the function's behavior at specific x-values. Let's examine the table:

Each row represents a coordinate point (x, g(x)) on the graph of the function. By plotting these points, we can visualize the function's shape and trend. When x = -3, g(x) = -1. This confirms that the function is negative for x values less than -2, as expected. Using the simplified form, g(-3) = (-3 + 2)³ = (-1)³ = -1. When x = -2, g(x) = 0. This verifies our earlier calculation of the root of the function. g(-2) = (-2 + 2)³ = 0³ = 0. When x = 0, g(x) = 8. This gives us the y-intercept of the function. g(0) = (0 + 2)³ = 2³ = 8. When x = 2, g(x) = 64. This demonstrates the rapid increase in the function's value as x moves away from the root. g(2) = (2 + 2)³ = 4³ = 64. When x = 3, g(x) = 125. This further reinforces the increasing trend of the function. g(3) = (3 + 2)³ = 5³ = 125. The table clearly shows the monotonic increasing behavior of the function. As x increases, g(x) also increases. The rate of increase is not constant, indicating the cubic nature of the function. The function grows more rapidly as x moves further away from the root at x = -2.

Determining the Function's Value for any Input

One of the key strengths of understanding a function's algebraic form is the ability to determine its value for any given input. Whether we have the original cubic form or the simplified form, we can substitute any value for x and calculate the corresponding value of g(x). Let's consider an example. Suppose we want to find the value of g(1). Using the simplified form, we have:

g(1) = (1 + 2)³ = 3³ = 27

This means that when x = 1, the function's value is 27. We can verify this using the original cubic form as well:

g(1) = 1³ + 6(1)² + 12(1) + 8 = 1 + 6 + 12 + 8 = 27

The result is the same, demonstrating the equivalence of the two forms. Another example: Let's find g(-1):

g(-1) = (-1 + 2)³ = 1³ = 1

Or, using the cubic form:

g(-1) = (-1)³ + 6(-1)² + 12(-1) + 8 = -1 + 6 - 12 + 8 = 1

We can also use the table to estimate function values that are not directly listed. For instance, if we wanted to approximate g(1.5), we could interpolate between the values of g(0) = 8 and g(2) = 64. However, it's important to remember that interpolation provides an approximation, and the accuracy depends on the function's behavior in that interval. In this case, due to the cubic nature of the function, linear interpolation might not be highly accurate. It is always better to use the function's equation to find the exact value.

Conclusion

In this exploration, we have delved into the function g(x) = x³ + 6x² + 12x + 8, uncovering its algebraic structure and analyzing its behavior using a table of values. We recognized the function as a cubic polynomial and, more importantly, simplified it to the form (x + 2)³. This simplification provided insights into the function's root and its monotonic increasing behavior. The table of values confirmed these observations and allowed us to visualize the function's trend. We also demonstrated how to determine the function's value for any input by substituting into the algebraic form, highlighting the power of mathematical expressions in predicting and understanding function behavior. By combining algebraic analysis with numerical evaluation, we have gained a comprehensive understanding of the function g(x). This approach is applicable to a wide range of functions and is a cornerstone of mathematical analysis.