What Are The Domain And Range Of F(x)=(1/5)^x? Explain The Domain And Range In Detail.

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In the realm of mathematics, understanding the domain and range of a function is crucial for grasping its behavior and characteristics. Specifically, when dealing with exponential functions, this knowledge becomes even more vital. Let's delve into the function f(x) = (1/5)^x and explore its domain and range in detail.

Domain of f(x) = (1/5)^x

When we discuss the domain of a function, we are essentially asking: for what input values (x-values) is the function defined? In other words, what values can we plug into the function and get a valid output? For the exponential function f(x) = (1/5)^x, we need to consider if there are any restrictions on the values we can input for x. Exponential functions, in their basic form, do not have any inherent restrictions on their domain. We can raise a positive number (in this case, 1/5) to any real power. Consider these scenarios:

  • Positive values of x: If we plug in a positive value for x, say x = 2, we get f(2) = (1/5)^2 = 1/25, a perfectly valid output.
  • Negative values of x: If we plug in a negative value for x, say x = -1, we get f(-1) = (1/5)^(-1) = 5, which is also a valid output. Negative exponents simply mean we take the reciprocal of the base raised to the positive exponent.
  • Zero: If we plug in x = 0, we get f(0) = (1/5)^0 = 1, as any non-zero number raised to the power of 0 is 1.
  • Fractions and irrational numbers: We can also use fractional or irrational numbers as exponents. For instance, f(1/2) = (1/5)^(1/2) represents the square root of 1/5, which is a real number. Similarly, we can evaluate f(π) = (1/5)^π using the properties of exponents and logarithms.

Given these observations, it becomes evident that we can input any real number for x in the function f(x) = (1/5)^x and obtain a valid real number output. Therefore, the domain of the function f(x) = (1/5)^x is all real numbers. We can represent this mathematically as:

Domain: (-∞, ∞)

This notation signifies that the domain includes all numbers from negative infinity to positive infinity, encompassing the entire real number line. This broad domain is a characteristic feature of exponential functions where the base is a positive number.

Range of f(x) = (1/5)^x

Now, let's shift our focus to the range of the function. The range represents the set of all possible output values (y-values) that the function can produce. To determine the range of f(x) = (1/5)^x, we need to analyze how the function behaves as x takes on different values.

  • Positive Outputs: Since the base of the exponential function (1/5) is a positive number, raising it to any real power will always result in a positive output. A positive number raised to any power, whether positive, negative, or zero, cannot be negative or zero. For example:
    • When x is a large positive number, (1/5)^x becomes a very small positive number, approaching zero but never actually reaching it.
    • When x is a large negative number, (1/5)^x becomes a very large positive number.
  • Asymptotic Behavior: As x approaches positive infinity, the value of f(x) = (1/5)^x gets closer and closer to zero, but it never actually reaches zero. This behavior is known as asymptotic behavior, and the x-axis (y = 0) acts as a horizontal asymptote for the function. This means the graph of the function approaches the x-axis but never intersects it.
  • Never Zero or Negative: The function will never output zero because no matter what power we raise 1/5 to, the result will always be a positive number. Similarly, the output will never be negative.

Considering these characteristics, we can conclude that the range of the function f(x) = (1/5)^x consists of all positive real numbers. The function can take on any positive value, from infinitesimally close to zero to infinitely large values, but it will never be zero or negative. Therefore, the range of the function f(x) = (1/5)^x is all real numbers greater than zero. We can express this mathematically as:

Range: (0, ∞)

This notation indicates that the range includes all numbers greater than zero, extending to positive infinity, but excluding zero itself. This is a typical range for exponential functions with a positive base that is not equal to 1.

Graphical Representation

Visualizing the graph of f(x) = (1/5)^x can further solidify our understanding of its domain and range. The graph is a decreasing exponential curve that approaches the x-axis (y = 0) as x increases, but never touches it. It starts from very large positive y-values when x is a large negative number and gradually decreases towards zero as x increases. The graph confirms that the function is defined for all real numbers (domain) and its output is always positive (range).

Conclusion

In summary, for the exponential function f(x) = (1/5)^x:

  • The domain is all real numbers, represented as (-∞, ∞).
  • The range is all real numbers greater than zero, represented as (0, ∞).

Understanding the domain and range of exponential functions is fundamental in mathematics, as it allows us to predict the function's behavior and interpret its results accurately. By recognizing the unrestricted input values (domain) and the exclusively positive output values (range) of f(x) = (1/5)^x, we gain a comprehensive understanding of this particular exponential function and exponential functions in general. This knowledge is invaluable in various mathematical applications, including calculus, differential equations, and mathematical modeling.

By mastering the concepts of domain and range, we unlock deeper insights into the world of functions and their applications. The ability to identify the domain and range is not just a mathematical exercise; it's a critical skill that enables us to analyze, interpret, and apply mathematical models to real-world scenarios with confidence and precision.