Finding K For Functions G(x) = X - 8 And H(x) = X / (3x - 2)

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This article explores the functions g(x) = x - 8 and h(x) = x / (3x - 2), with a particular focus on determining the value of k for which the function h(x) is not defined. We'll delve into the concepts of functions, their domains, and how to identify restrictions on the input values. Understanding these concepts is crucial for mastering advanced mathematical topics and solving complex problems. We will break down the process of finding the value of k step-by-step, ensuring a clear understanding of the underlying principles.

Defining the Functions

Before we dive into the specifics of finding the value of k, let's clearly define the functions given:

  • Function g(x): g(x) = x - 8. This is a simple linear function. For any input value x, the function subtracts 8 from it. Linear functions like this are straightforward and have a domain of all real numbers, meaning there are no restrictions on the values we can input for x.
  • Function h(x): h(x) = x / (3x - 2), x ≠ k. This is a rational function, which means it's a fraction where both the numerator and the denominator are polynomials. The presence of x in the denominator introduces a critical restriction: the denominator cannot be equal to zero. This is because division by zero is undefined in mathematics. The condition x ≠ k indicates that there's a specific value k that x cannot take, as it would make the denominator zero. Our primary goal is to find this value of k. Rational functions are fundamental in calculus and other advanced mathematical areas, so mastering their behavior is essential.

The Significance of k and the Domain of h(x)

The value k represents a crucial point in the context of the function h(x). It's the value that, if substituted for x, would make the denominator of the fraction equal to zero. As we know, division by zero is undefined in mathematics, rendering the function h(x) meaningless at x = k. Therefore, k represents a restriction on the domain of h(x). The domain of a function is the set of all possible input values (x values) for which the function produces a valid output. In the case of h(x), the domain consists of all real numbers except for k. Identifying such restrictions is a vital part of understanding the behavior and limitations of a given function.

Why is it so important to exclude k from the domain? Because if we were to graph the function h(x), the value x = k would correspond to a vertical asymptote. A vertical asymptote is a vertical line that the graph of the function approaches but never actually touches. This signifies a point where the function's output grows infinitely large (either positively or negatively) as x gets closer and closer to k. In practical applications, understanding these domain restrictions is crucial for avoiding errors and misinterpretations when using mathematical models.

Finding the Value of k

To determine the value of k, we need to find the value of x that makes the denominator of h(x) equal to zero. The denominator of h(x) is 3x - 2. Therefore, we need to solve the following equation:

3x - 2 = 0

This is a simple linear equation. To solve for x, we can follow these steps:

  1. Add 2 to both sides of the equation:

    3x = 2

  2. Divide both sides by 3:

    x = 2/3

Therefore, the value of k is 2/3. This means that the function h(x) = x / (3x - 2) is undefined when x = 2/3. In other words, the domain of h(x) is all real numbers except for 2/3. This is a critical piece of information when working with this function, as substituting x = 2/3 into h(x) will result in division by zero, which is an invalid operation.

Implications and Further Exploration

Finding the value of k not only tells us where the function h(x) is undefined but also provides insights into its overall behavior. As x approaches 2/3, the value of h(x) will either approach positive infinity or negative infinity. This behavior is characteristic of rational functions near their vertical asymptotes. Understanding these asymptotes is essential for sketching the graph of h(x) and analyzing its properties.

Further exploration of these functions could involve:

  • Graphing both g(x) and h(x) to visually represent their behavior.
  • Analyzing the range of h(x), which is the set of all possible output values.
  • Investigating the limits of h(x) as x approaches infinity and negative infinity.
  • Determining the inverse function of g(x) and h(x), if they exist.

By delving deeper into these aspects, we can gain a more comprehensive understanding of the properties and applications of these functions.

Conclusion

In summary, we successfully identified the value of k for the function h(x) = x / (3x - 2) by setting the denominator equal to zero and solving for x. We found that k = 2/3, which means that h(x) is undefined when x = 2/3. This value represents a critical restriction on the domain of h(x) and corresponds to a vertical asymptote on its graph. Understanding how to find such restrictions is a fundamental skill in mathematics, particularly when dealing with rational functions. The process we followed highlights the importance of identifying potential issues like division by zero and taking them into account when working with mathematical functions. This analysis not only solves the specific problem but also provides a solid foundation for understanding more complex mathematical concepts and applications.