Defining Function H(x) Exploring Relationships Between Functions F, G, And H
In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships and processes. To truly grasp their essence, we must delve into their definitions, explore their properties, and understand how they interact with one another. In this comprehensive exploration, we will consider three functions, f, g, and h, each defined by a distinct algebraic expression. Our goal is to unravel the expression that precisely defines function h, unraveling the intricate connections between these mathematical entities.
Defining the Functions: f(x), g(x), and h(x)
Let's begin by laying the groundwork, meticulously defining the functions at the heart of our investigation.
Function f(x): A Cubic Polynomial
The first function, denoted as f(x), is a cubic polynomial, a mathematical expression characterized by the highest power of the variable x being 3. The specific expression that defines f(x) is:
f(x) = 3x³ + 9x² - 12x
This cubic polynomial encompasses a range of terms, each contributing to the overall behavior of the function. The term 3x³ signifies the cubic component, dictating the function's long-term trend as x approaches positive or negative infinity. The term 9x² represents the quadratic component, influencing the function's curvature and potential turning points. Finally, the term -12x embodies the linear component, contributing to the function's slope and intercept.
Function g(x): A Linear Expression
Moving on, the second function, symbolized as g(x), takes the form of a linear expression, a mathematical construct where the highest power of the variable x is 1. The explicit expression that defines g(x) is:
g(x) = x - 1
This linear expression represents a straight line when plotted on a graph. The coefficient of x, which is 1 in this case, dictates the slope of the line, indicating its steepness and direction. The constant term, -1, signifies the y-intercept, the point where the line intersects the vertical axis.
Function h(x): A Quadratic Expression
Finally, the third function, denoted as h(x), presents itself as a quadratic expression, a mathematical formulation where the highest power of the variable x is 2. The precise expression that defines h(x) is:
h(x) = 3x² + 12x
This quadratic expression represents a parabola when plotted on a graph. The coefficient of x², which is 3 in this case, determines the parabola's concavity, indicating whether it opens upwards or downwards. The coefficient of x, which is 12, influences the parabola's position and orientation.
Unveiling the Relationship: Defining h(x) in Terms of f(x) and g(x)
Now, the central question beckons: Which expression accurately defines function h(x) in relation to functions f(x) and g(x)? To embark on this quest, we must explore the potential connections between these functions, examining various mathematical operations and their implications.
The options presented for defining h(x) are:
- (A) (f - g)(x)
- (B) (f · g)(x)
Let's dissect each option meticulously, unraveling their mathematical essence and determining their compatibility with the given functions.
Option A: (f - g)(x) - The Difference of Functions
The expression (f - g)(x) signifies the difference between functions f(x) and g(x). To evaluate this expression, we subtract the expression for g(x) from the expression for f(x). Let's perform this operation step by step:
(f - g)(x) = f(x) - g(x)
Substitute the expressions for f(x) and g(x):
(f - g)(x) = (3x³ + 9x² - 12x) - (x - 1)
Distribute the negative sign:
(f - g)(x) = 3x³ + 9x² - 12x - x + 1
Combine like terms:
(f - g)(x) = 3x³ + 9x² - 13x + 1
Comparing this result with the expression for h(x), which is 3x² + 12x, we observe a stark discrepancy. The expression (f - g)(x) yields a cubic polynomial, while h(x) is a quadratic expression. Therefore, option A, (f - g)(x), does not define function h(x).
Option B: (f · g)(x) - The Product of Functions
The expression (f · g)(x) represents the product of functions f(x) and g(x). To evaluate this expression, we multiply the expression for f(x) by the expression for g(x). Let's execute this operation meticulously:
(f · g)(x) = f(x) · g(x)
Substitute the expressions for f(x) and g(x):
(f · g)(x) = (3x³ + 9x² - 12x) · (x - 1)
Distribute the terms:
(f · g)(x) = 3x⁴ + 9x³ - 12x² - 3x³ - 9x² + 12x
Combine like terms:
(f · g)(x) = 3x⁴ + 6x³ - 21x² + 12x
Upon comparing this result with the expression for h(x), which is 3x² + 12x, we again encounter a significant mismatch. The expression (f · g)(x) results in a quartic polynomial, while h(x) is a quadratic expression. Consequently, option B, (f · g)(x), does not define function h(x) either.
An Alternative Path: Factoring and Unveiling the Connection
Since the provided options do not directly define h(x), let's explore an alternative approach: factoring the expression for f(x) and seeking a potential connection with g(x).
Begin by factoring out the greatest common factor from f(x):
f(x) = 3x³ + 9x² - 12x = 3x(x² + 3x - 4)
Now, factor the quadratic expression within the parentheses:
f(x) = 3x(x + 4)(x - 1)
Notice the factor (x - 1) within the factored expression for f(x). This factor is precisely the expression for g(x). Let's rewrite f(x) to highlight this connection:
f(x) = 3x(x + 4)g(x)
Now, let's consider the expression 3x² + 12x, which defines h(x). We can factor out 3x from this expression:
h(x) = 3x² + 12x = 3x(x + 4)
Comparing this factored expression for h(x) with the factored expression for f(x), we observe a striking similarity. The expression 3x(x + 4) appears in both.
The Definitive Expression for h(x): A Quotient of Functions
Based on our factoring analysis, we can deduce that h(x) can be defined as the quotient of f(x) and g(x), with a crucial adjustment:
h(x) = f(x) / g(x) = [3x(x + 4)(x - 1)] / (x - 1)
By canceling the common factor of (x - 1), we arrive at:
h(x) = 3x(x + 4) = 3x² + 12x
This expression precisely matches the given definition of h(x). Therefore, we can definitively state that h(x) is defined as the quotient of f(x) and g(x), with the common factor of (x - 1) carefully considered.
Conclusion: Unraveling the Interconnectedness of Functions
In this comprehensive exploration, we embarked on a journey to unveil the expression that defines function h(x) in relation to functions f(x) and g(x). We meticulously analyzed the given functions, explored potential relationships through mathematical operations, and ultimately discovered that h(x) can be defined as the quotient of f(x) and g(x), with the common factor of (x - 1) appropriately addressed.
This endeavor underscores the interconnectedness of functions in mathematics. By understanding their definitions, properties, and potential interactions, we can unravel complex relationships and gain deeper insights into the mathematical world. The ability to manipulate functions, factor expressions, and identify common factors empowers us to solve intricate problems and appreciate the elegance of mathematical structures.
This exploration serves as a testament to the power of mathematical reasoning and the beauty of functional relationships. As we continue our mathematical journey, may we embrace the challenge of unraveling complex connections and celebrate the profound insights that mathematics offers.
Which of the following expressions defines the function h(x)?
Defining Function h(x) Exploring Relationships Between Functions f, g, and h
