Evaluating Functions F(x), G(x), And H(x) A Step By Step Guide

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In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding various relationships. This article delves into three distinct functions – f(x), g(x), and h(x) – exploring their definitions, evaluations, and significance. We will meticulously calculate the values of these functions at specific points, providing a comprehensive understanding of their behavior.

Defining the Functions

Before we embark on evaluating these functions, let's first define them explicitly:

  • Function f(x): This function is defined as $f(x) = rac{x^3 + 2}{x^3}$. It involves a rational expression where the numerator is the sum of the cube of the input variable 'x' and the constant 2, while the denominator is simply the cube of 'x'. This function exhibits interesting characteristics, particularly as 'x' approaches zero or infinity.
  • Function g(x): The function g(x) is defined as $g(x) =

\sqrt{9 + 3x}$. This function involves a square root, indicating that the input variable 'x' must satisfy the condition 9 + 3x ≥ 0 for the function to be defined in the real number system. The square root introduces a non-linear relationship between 'x' and the output of the function.

  • Function h(x): The function h(x) is defined as $h(x) = |4x - 8|$. This function incorporates the absolute value, which means the output will always be non-negative, regardless of the sign of the expression 4x - 8. The absolute value introduces a piecewise nature to the function, as its behavior changes depending on whether 4x - 8 is positive or negative.

Evaluating Function f(x) at x = 4

Let's start by evaluating the function f(x) at x = 4. This involves substituting '4' for 'x' in the function's definition:

f(4) = rac{4^3 + 2}{4^3}

Now, we simplify the expression:

f(4) = rac{64 + 2}{64}

f(4) = rac{66}{64}

We can further simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

f(4) = rac{33}{32}

Therefore, the value of the function f(x) at x = 4 is 33/32. This result indicates that when the input to the function f(x) is 4, the output is slightly greater than 1. This is due to the fact that the numerator (66) is slightly larger than the denominator (64). Understanding this evaluation helps us grasp the function's behavior around the point x = 4.

Evaluating Function g(x) at x = 2

Next, we will evaluate the function g(x) at x = 2. Again, we substitute '2' for 'x' in the function's definition:

g(2)=9+3(2)g(2) = \sqrt{9 + 3(2)}

Now, we simplify the expression under the square root:

g(2)=9+6g(2) = \sqrt{9 + 6}

g(2)=15g(2) = \sqrt{15}

The square root of 15 cannot be simplified further into integers, as 15 does not have any perfect square factors other than 1. Therefore, the value of the function g(x) at x = 2 is √15. This result signifies that when the input to the function g(x) is 2, the output is an irrational number approximately equal to 3.87. This evaluation highlights the impact of the square root function in generating non-integer outputs.

Evaluating Function h(x) at x = -1/2

Finally, let's evaluate the function h(x) at x = -1/2. We substitute '-1/2' for 'x' in the function's definition:

h(−12)=∣4(−12)−8∣h(-\frac{1}{2}) = |4(-\frac{1}{2}) - 8|

Now, we simplify the expression inside the absolute value:

h(−12)=∣−2−8∣h(-\frac{1}{2}) = |-2 - 8|

h(−12)=∣−10∣h(-\frac{1}{2}) = |-10|

The absolute value of -10 is 10. Therefore, the value of the function h(x) at x = -1/2 is 10. This result demonstrates the effect of the absolute value function, which transforms any negative value into its positive counterpart. In this case, the expression 4x - 8 evaluates to -10 when x is -1/2, but the absolute value ensures that the final output is 10.

Summary of Evaluations

In summary, we have evaluated the three functions at the specified points:

  • f(4) = 33/32
  • g(2) = √15
  • h(-1/2) = 10

These evaluations provide insights into the behavior of each function at specific input values. Function f(x) produces a rational output slightly greater than 1 at x = 4. Function g(x) yields an irrational output at x = 2, demonstrating the nature of square root functions. Function h(x), due to the absolute value, produces a positive output even for a negative input value.

Significance of Function Evaluation

Evaluating functions is a cornerstone of mathematical analysis. It allows us to: understand how a function behaves at different points, visualize the function's graph, identify key features like intercepts and extrema, and build mathematical models for real-world phenomena. By calculating function values, we bridge the gap between abstract mathematical definitions and concrete numerical results, facilitating a deeper comprehension of functional relationships.

Moreover, function evaluation plays a crucial role in various applications, such as: data analysis, where functions are used to model trends and patterns; computer programming, where functions are fundamental building blocks of algorithms; and engineering, where functions are used to describe physical systems and processes. The ability to accurately evaluate functions is therefore an essential skill for anyone working with mathematical models and quantitative data.

Further Exploration of Functions

This exploration of functions f(x), g(x), and h(x) serves as a foundation for deeper investigations. We can further analyze these functions by: determining their domains and ranges, graphing their behavior, finding their derivatives and integrals (if applicable), and exploring their applications in various contexts. By delving into these aspects, we can gain a more complete understanding of the characteristics and significance of these functions within the broader landscape of mathematics.

In conclusion, understanding and evaluating functions is a critical skill in mathematics. By meticulously analyzing the definitions of f(x), g(x), and h(x), and by calculating their values at specific points, we have gained valuable insights into their behavior and applications. This exploration paves the way for further analysis and a deeper appreciation of the power and versatility of functions in mathematical modeling and problem-solving.