Domain Range And Asymptote Of H(x) = (0.5)^x - 9 Explained

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In the realm of mathematics, understanding the characteristics of functions is paramount. Functions serve as the fundamental building blocks for modeling real-world phenomena, and their behavior dictates the outcomes of various processes. Among the myriad types of functions, exponential functions hold a special place due to their ability to describe phenomena involving rapid growth or decay. In this comprehensive exploration, we delve into the intricacies of the exponential function h(x) = (0.5)^x - 9, dissecting its domain, range, and asymptote, which are crucial for comprehending its behavior and applications.

Decoding the Domain: Unveiling the Function's Input Territory

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the collection of x-values that can be plugged into the function without causing any mathematical mishaps, such as division by zero or taking the square root of a negative number. For the function h(x) = (0.5)^x - 9, we need to examine the expression (0.5)^x to determine the domain.

Exponential functions, in their purest form, have a base raised to a variable exponent. The base can be any positive real number, except for 1, as 1 raised to any power always results in 1, making it a rather uninteresting case. In our function, the base is 0.5, which is indeed a positive real number different from 1. The exponent, x, can be any real number without causing any mathematical contradictions. We can plug in positive numbers, negative numbers, zero, fractions, or even irrational numbers like pi, and the expression (0.5)^x will still yield a real number.

Therefore, the domain of h(x) = (0.5)^x - 9 is the set of all real numbers. This can be expressed mathematically as:

Domain: {x | x is a real number}

This means that there are no restrictions on the input values for this function. We can substitute any real number for x, and the function will produce a corresponding output value.

Mapping the Range: Tracing the Function's Output Landscape

The range of a function is the set of all possible output values (y-values) that the function can produce. In essence, it's the collection of all the results we can obtain by plugging in the various values from the domain. To determine the range of h(x) = (0.5)^x - 9, we need to analyze the behavior of the exponential term (0.5)^x and how it affects the overall output of the function.

The exponential term (0.5)^x is a decreasing exponential function. As x increases, the value of (0.5)^x decreases, approaching 0 but never actually reaching it. This is because any positive number raised to a negative power will always be greater than 0. Conversely, as x decreases (becomes more negative), the value of (0.5)^x increases without bound, getting larger and larger. Think of it as repeatedly multiplying 0.5 by itself as x becomes a large negative number – the result will be a very large number.

The subtraction of 9 from the exponential term (0.5)^x shifts the entire function downward by 9 units. This means that the output values of the function will be 9 units lower than the output values of the basic exponential function (0.5)^x. Since (0.5)^x can take on any positive value (approaching 0 but never reaching it), subtracting 9 from it means that the function h(x) = (0.5)^x - 9 can take on any value greater than -9 but will never actually reach -9.

Therefore, the range of h(x) = (0.5)^x - 9 is the set of all real numbers greater than -9. Mathematically, this can be expressed as:

Range: {y | y > -9}

This signifies that the output values of the function will always be greater than -9, but the function will never actually produce the value -9.

Pinpointing the Asymptote: Identifying the Function's Boundary Line

An asymptote is a line that a curve approaches as it heads towards infinity. It's like a boundary line that the function gets closer and closer to but never quite touches or crosses. Asymptotes provide valuable insights into the long-term behavior of a function, indicating where the function is headed as the input values become extremely large or small. For the function h(x) = (0.5)^x - 9, we're interested in the horizontal asymptote, which is a horizontal line that the function approaches as x approaches positive or negative infinity.

To determine the horizontal asymptote, we need to examine the behavior of the function as x approaches positive and negative infinity. As we discussed earlier, as x increases (approaches positive infinity), the value of (0.5)^x decreases, approaching 0. Consequently, the value of h(x) = (0.5)^x - 9 approaches 0 - 9, which is -9.

As x decreases (approaches negative infinity), the value of (0.5)^x increases without bound. However, the subtraction of 9 from (0.5)^x does not change the fact that the function will continue to increase without bound as x becomes increasingly negative. The function will not approach any particular horizontal line as x approaches negative infinity.

Therefore, the function h(x) = (0.5)^x - 9 has a horizontal asymptote at the line y = -9. This means that as x approaches positive infinity, the function gets closer and closer to the line y = -9 but never actually touches or crosses it. The line y = -9 serves as a lower boundary for the function's output values.

Asymptote: y = -9

Summarizing the Key Characteristics

In conclusion, the exponential function h(x) = (0.5)^x - 9 exhibits the following key characteristics:

  • Domain: {x | x is a real number}
  • Range: {y | y > -9}
  • Asymptote: y = -9

Understanding these characteristics provides a comprehensive picture of the function's behavior. The domain tells us the allowed input values, the range reveals the possible output values, and the asymptote indicates the function's long-term trend as the input values become extremely large or small. These insights are essential for effectively utilizing exponential functions in mathematical modeling and real-world applications.

By meticulously analyzing the domain, range, and asymptote of h(x) = (0.5)^x - 9, we've gained a deeper appreciation for the intricacies of exponential functions and their role in describing phenomena involving exponential decay. This exploration serves as a foundation for further investigations into the world of functions and their applications in various fields of study.