Domain And Range Of Exponential Function Analysis From A Table

Understanding Exponential Functions

Before we dive into determining the domain and range of the given exponential function, let's first establish a solid understanding of what exponential functions are and their key characteristics. An exponential function is a mathematical function in which the independent variable (typically x) appears in the exponent. The general form of an exponential function is f(x) = abˣ, where a is the initial value (the value of the function when x is 0), b is the base (a positive real number not equal to 1), and x is the exponent. Exponential functions exhibit a rapid rate of change, either increasing (exponential growth) or decreasing (exponential decay) as the independent variable changes. The base b determines whether the function represents growth (b > 1) or decay (0 < b < 1). Understanding these fundamentals is crucial for analyzing and interpreting exponential functions, including determining their domain and range.

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For exponential functions, the domain is typically all real numbers, meaning that you can plug in any real number for x and get a valid output. This is because there are no restrictions on the values that x can take in the exponent. You can raise a positive base to any power, whether it's positive, negative, zero, or a fraction. However, there can be situations where the domain is restricted based on the context of the problem or the specific application of the function. For instance, if the exponential function models a real-world scenario like population growth, the domain might be limited to non-negative values since time cannot be negative. In mathematical terms, the domain of an exponential function is often expressed as (-∞, ∞), representing all real numbers.

The range of a function refers to the set of all possible output values (y-values) that the function can produce. For exponential functions of the form f(x) = abˣ, where a is a non-zero constant and b is a positive base not equal to 1, the range is determined by the value of a and whether the function represents exponential growth or decay. If a is positive and b > 1 (exponential growth), the range is (0, ∞), meaning the function can take on any positive value but never reaches zero. If a is positive and 0 < b < 1 (exponential decay), the range is also (0, ∞). The function approaches zero as x increases but never actually reaches it. If a is negative, the range is reflected across the x-axis, becoming (-∞, 0) for both growth and decay scenarios. In essence, the range of a basic exponential function is always a set of positive or negative values, depending on the sign of the coefficient a, and it never includes zero.

Analyzing the Table

To determine the domain and range of the function represented by the given table, we must first identify the type of function and its key characteristics. The table provides a set of ordered pairs (x, y) that represent points on the graph of the function. By observing the y-values, we can see a pattern of growth. As x increases, the y-values also increase, suggesting an exponential relationship. Specifically, each y-value appears to be multiplied by a constant factor to obtain the next y-value. This constant factor is crucial for identifying the base of the exponential function. To find the base, we can divide any y-value by its preceding y-value. For example, 5 / 4 = 1.25, 6.25 / 5 = 1.25, and 7.8125 / 6.25 = 1.25. This consistent ratio of 1.25 indicates that the function is exponential with a base of 1.25. Recognizing this pattern is essential for further analysis of the function's domain and range.

Now that we've identified the function as exponential with a base of 1.25, we can determine the initial value (a) by looking at the y-value when x = 0. From the table, we see that when x = 0, y = 4. Therefore, the initial value a is 4. This means the function can be represented in the form f(x) = 4(1.25)ˣ. Having the explicit form of the function allows us to analyze its domain and range more effectively. The initial value plays a significant role in determining the range, as it represents the y-intercept of the function and influences the overall behavior of the exponential growth or decay. In this case, since the initial value is positive and the base is greater than 1, the function represents exponential growth starting from 4.

With the exponential function identified as f(x) = 4(1.25)ˣ, we can now analyze its domain and range. The domain of an exponential function is the set of all possible x-values that can be input into the function. In this case, since we are dealing with a continuous exponential function, there are no restrictions on the x-values. We can input any real number for x, and the function will produce a valid output. Therefore, the domain of the function is all real numbers, which can be expressed in interval notation as (-∞, ∞). Understanding the concept of domain is crucial in mathematics as it defines the boundaries within which a function operates and provides context for its behavior.

Determining the Domain and Range

The range, on the other hand, represents the set of all possible y-values that the function can output. For an exponential function of the form f(x) = abˣ, the range depends on the sign of a and the base b. In our case, a is 4 (positive) and b is 1.25 (greater than 1), indicating exponential growth. This means the function will always produce positive y-values. As x approaches negative infinity, the function approaches 0 but never actually reaches it. As x approaches positive infinity, the function grows without bound. Therefore, the range of the function is all positive real numbers, which can be expressed in interval notation as (0, ∞). The range provides valuable insight into the output values that a function can generate and is essential for understanding its overall behavior and limitations.

Conclusion

In conclusion, for the continuous exponential function represented by the table, the domain is all real numbers (-∞, ∞), and the range is all positive real numbers (0, ∞). Understanding the domain and range of functions is fundamental in mathematics, as it helps us to define the boundaries and limitations of functions, interpret their behavior, and apply them effectively in various contexts. In the case of exponential functions, the domain typically encompasses all real numbers, while the range is influenced by the initial value and the base, determining whether the function exhibits exponential growth or decay and the set of possible output values.

In summary:

• Domain: (-∞, ∞) or all real numbers
• Range: (0, ∞) or all positive real numbers

What are the domain and range of the exponential function represented by the table?

Domain and Range of Exponential Function Analysis From a Table