How To Evaluate The Piecewise Function H(x) At X=3, X=0, And X=6?

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Piecewise functions, a fascinating concept in mathematics, are functions defined by multiple sub-functions, each applying to a specific interval of the independent variable. Understanding how to evaluate piecewise functions is crucial for various mathematical applications. This article delves into the intricacies of piecewise function evaluation, providing a step-by-step approach with examples to solidify your understanding. We will specifically address how to evaluate the given piecewise function at different values of the independent variable.

Understanding Piecewise Functions

Before diving into the evaluation process, let's first grasp the essence of piecewise functions. A piecewise function is essentially a function composed of multiple sub-functions, each with its own domain. The domain dictates the input values for which a particular sub-function is applicable. These sub-functions piece together to define the overall behavior of the function across its entire domain.

The key to working with piecewise functions lies in identifying the correct sub-function to use for a given input value. This is determined by checking which interval the input value falls into. Each sub-function is associated with a specific interval, and only the sub-function corresponding to the interval containing the input value should be used for evaluation.

Piecewise functions are not just abstract mathematical constructs; they have real-world applications in various fields. For instance, they can model tax brackets, where the tax rate varies depending on income level. They can also represent the cost of services that change based on usage, such as electricity bills with tiered pricing. Understanding piecewise functions provides a powerful tool for modeling and analyzing situations with varying conditions.

Step-by-Step Guide to Evaluating Piecewise Functions

To effectively evaluate a piecewise function, follow these simple yet crucial steps:

  1. Identify the Input Value: The first step is to clearly identify the input value, the value for which you want to evaluate the function. This is the value that will be substituted into the appropriate sub-function.
  2. Determine the Relevant Interval: Next, determine which interval the input value belongs to. This is the most critical step, as it dictates which sub-function you'll use. Carefully examine the conditions defining each interval and compare the input value to these conditions.
  3. Select the Corresponding Sub-function: Once you've identified the interval, select the corresponding sub-function associated with that interval. This is the sub-function that will be used to calculate the output value.
  4. Substitute and Evaluate: Substitute the input value into the selected sub-function. Perform the necessary calculations according to the sub-function's formula. The result of this calculation is the function's output value for the given input.

By following these steps meticulously, you can confidently evaluate piecewise functions for any input value. Remember, the key is to correctly identify the interval and the corresponding sub-function.

Evaluating the Given Piecewise Function

Let's apply the steps outlined above to evaluate the given piecewise function:

 h(x) = { (x^2 - 36) / (x - 6)   if x ≠ 6
         { 8                     if x = 6

This function, denoted as h(x), is defined by two sub-functions. The first sub-function, (x^2 - 36) / (x - 6), applies when x is not equal to 6. The second sub-function, 8, applies specifically when x is equal to 6. This clear distinction is crucial for accurate evaluation. Now, let's evaluate this function for the given input values.

(a) Evaluating h(3)

  1. Identify the Input Value: The input value is 3.

  2. Determine the Relevant Interval: Since 3 is not equal to 6, it falls into the interval defined by the condition x ≠ 6.

  3. Select the Corresponding Sub-function: The sub-function corresponding to x ≠ 6 is (x^2 - 36) / (x - 6).

  4. Substitute and Evaluate: Substitute x = 3 into the sub-function:

    h(3) = (3^2 - 36) / (3 - 6) = (9 - 36) / (-3) = -27 / -3 = 9

    Therefore, h(3) = 9.

(b) Evaluating h(0)

  1. Identify the Input Value: The input value is 0.

  2. Determine the Relevant Interval: Since 0 is not equal to 6, it falls into the interval defined by the condition x ≠ 6.

  3. Select the Corresponding Sub-function: The sub-function corresponding to x ≠ 6 is (x^2 - 36) / (x - 6).

  4. Substitute and Evaluate: Substitute x = 0 into the sub-function:

    h(0) = (0^2 - 36) / (0 - 6) = (-36) / (-6) = 6

    Therefore, h(0) = 6.

(c) Evaluating h(6)

  1. Identify the Input Value: The input value is 6.

  2. Determine the Relevant Interval: Since the input value is exactly 6, it falls into the interval defined by the condition x = 6.

  3. Select the Corresponding Sub-function: The sub-function corresponding to x = 6 is 8.

  4. Substitute and Evaluate: In this case, the sub-function is a constant value. There's no need for substitution; the value is directly given as:

    h(6) = 8

    Therefore, h(6) = 8.

Simplifying the Sub-function

Interestingly, the sub-function (x^2 - 36) / (x - 6) can be simplified. Recognizing this simplification can sometimes make evaluation easier and provide deeper insights into the function's behavior. The numerator, x^2 - 36, is a difference of squares, which can be factored as (x - 6)(x + 6). Thus, the sub-function can be rewritten as:

(x^2 - 36) / (x - 6) = ((x - 6)(x + 6)) / (x - 6)

For x ≠ 6, we can cancel out the (x - 6) terms, resulting in:

(x^2 - 36) / (x - 6) = x + 6

This simplification reveals that for all x values not equal to 6, the function behaves linearly, equivalent to the function x + 6. However, it's crucial to remember the original condition x ≠ 6. At x = 6, the original sub-function is undefined due to division by zero. This is why the piecewise function explicitly defines h(6) as 8, filling in this discontinuity. This technique of simplifying sub-functions, when possible, enhances our understanding of the piecewise function's nature and can streamline the evaluation process.

Conclusion

Evaluating piecewise functions requires careful attention to detail and a systematic approach. By identifying the correct interval for each input value and selecting the corresponding sub-function, you can accurately determine the function's output. This article has provided a comprehensive guide to evaluating piecewise functions, complete with step-by-step instructions and examples. Remember to always consider the conditions defining each interval and simplify sub-functions when possible to gain a deeper understanding of the function's behavior. Piecewise functions are a powerful tool in mathematics, and mastering their evaluation unlocks a wider range of problem-solving capabilities.

This detailed exploration should equip you with the necessary skills and knowledge to confidently tackle any piecewise function evaluation problem. Remember to practice consistently to solidify your understanding and enhance your problem-solving abilities.