Analyzing Exponential Function Y=750(1.4)^x Domain Range And Intercepts

In the realm of mathematics, exponential functions play a pivotal role in describing phenomena characterized by rapid growth or decay. These functions are ubiquitous in various scientific and real-world applications, including population dynamics, compound interest calculations, radioactive decay, and the spread of infectious diseases. Understanding the properties and behavior of exponential functions is, therefore, crucial for students, researchers, and professionals alike. This article aims to delve into the intricacies of a specific exponential function, $y = 750(1.4)^x$, by meticulously analyzing its key attributes. We will construct a comprehensive table that encapsulates the function's domain, range, intercepts, and other pertinent characteristics. By employing interval notation, we will precisely define the domain and range, providing a clear and concise representation of the function's boundaries. Furthermore, we will express intercepts as ordered pairs, adhering to standard mathematical conventions and facilitating ease of interpretation. In instances where an answer does not exist, we will denote it as "DNE," ensuring completeness and accuracy in our analysis. This exploration will not only enhance our understanding of the function $y = 750(1.4)^x$ but also provide a framework for analyzing other exponential functions, fostering a deeper appreciation for their mathematical significance.

Before we dive into the specifics of the function $y = 750(1.4)^x$, let's establish a foundational understanding of exponential functions in general. An exponential function is a mathematical function in the form $f(x) = ab^x$, where 'a' is the initial value or the y-intercept (when x = 0), 'b' is the base (a positive real number not equal to 1), and 'x' is the exponent. The base 'b' determines whether the function represents exponential growth (if b > 1) or exponential decay (if 0 < b < 1). The coefficient 'a' scales the function vertically, affecting the steepness of the curve. The exponent 'x' represents the independent variable, and its value directly influences the output of the function.

Exponential functions are characterized by their rapid rate of change. As 'x' increases, the function's value either grows very quickly (in the case of exponential growth) or diminishes rapidly towards zero (in the case of exponential decay). This behavior distinguishes exponential functions from linear functions, which have a constant rate of change. The domain of an exponential function, unless restricted by a specific context, is typically all real numbers, meaning that 'x' can take any real value. However, the range is usually limited to positive values (excluding zero) for exponential growth functions and positive values less than 'a' for exponential decay functions. The horizontal asymptote, a line that the function approaches but never touches, is the x-axis (y = 0) for most basic exponential functions.

Exponential functions are widely used in modeling various real-world phenomena. For example, the growth of a bacterial population can be modeled using an exponential function, where the base 'b' represents the growth rate and the coefficient 'a' represents the initial population size. Similarly, the decay of a radioactive substance can be modeled using an exponential function, where the base 'b' represents the decay rate and the coefficient 'a' represents the initial amount of the substance. In finance, compound interest is a classic example of exponential growth, where the base 'b' is (1 + interest rate) and the coefficient 'a' is the principal amount. Understanding the properties of exponential functions is, therefore, essential for making accurate predictions and informed decisions in various fields.

Now, let's focus on the specific exponential function at hand: $y = 750(1.4)^x$. This function is in the form $f(x) = ab^x$, where $a = 750$ and $b = 1.4$. Since the base $b = 1.4$ is greater than 1, this function represents exponential growth. This means that as the value of 'x' increases, the value of 'y' will increase rapidly. The coefficient $a = 750$ represents the initial value of the function, which is the value of 'y' when $x = 0$. This also corresponds to the y-intercept of the function.

To further analyze the function, we need to determine its domain, range, intercepts, and other key characteristics. The domain of an exponential function is typically all real numbers, unless there are specific restrictions imposed by the context of the problem. In this case, there are no such restrictions, so the domain is all real numbers. We can express this in interval notation as $(-\infty, \infty)$. The range of this function is the set of all positive real numbers, since the base is greater than 1 and the coefficient 'a' is positive. The function will approach the x-axis (y = 0) as 'x' decreases towards negative infinity, but it will never actually touch or cross the x-axis. Therefore, the range in interval notation is $(0, \infty)$.

To find the intercepts, we need to determine where the function intersects the x-axis and the y-axis. The y-intercept is the point where $x = 0$. Substituting $x = 0$ into the function, we get $y = 750(1.4)^0 = 750(1) = 750$. Therefore, the y-intercept is the ordered pair $(0, 750)$. The x-intercept is the point where $y = 0$. However, as we discussed earlier, this exponential function will never equal zero, so there is no x-intercept. We can denote this as DNE (Does Not Exist).

Now that we have analyzed the exponential function $y = 750(1.4)^x$ and determined its key characteristics, we can complete the table as requested. The table will include the domain, range, y-intercept, and x-intercept, with the domain and range expressed in interval notation and the intercepts expressed as ordered pairs. If an answer does not exist, we will write DNE.

Understanding the domain and range of an exponential function is crucial for comprehending its behavior and limitations. As we've established, the domain of $y = 750(1.4)^x$ is $(-\infty, \infty)$, signifying that any real number can be inputted as 'x'. This stems from the fundamental nature of exponentiation; we can raise a positive base (like 1.4) to any power, whether positive, negative, or zero, and the result will always be a real number.

However, the range is more constrained. For our function, the range is $(0, \infty)$, indicating that the output 'y' will always be a positive number. This is because 1.4 raised to any power will always be positive, and multiplying a positive number (750) by another positive number will invariably yield a positive result. Importantly, the function will never output zero or a negative value. This characteristic is a hallmark of exponential growth functions where the base is greater than 1.

To further illustrate this, consider what happens as 'x' becomes increasingly negative. While $1.4^x$ gets closer and closer to zero, it never actually reaches it. Multiplying by 750 doesn't change this; the function will asymptotically approach the x-axis (y = 0) but never intersect it. This concept of a horizontal asymptote is a key feature of exponential functions.

The absence of an x-intercept, which we previously denoted as DNE, is a direct consequence of this range restriction. Since the function never outputs zero, it can never cross the x-axis.

In contrast, the y-intercept, which we found to be (0, 750), represents the function's initial value. It's the point where the exponential curve starts its ascent. This value is directly determined by the coefficient 'a' in the general form $f(x) = ab^x$, which in our case is 750.

Intercepts are the points where a function's graph crosses the coordinate axes, providing valuable insights into the function's behavior. The y-intercept, as we've determined for $y = 750(1.4)^x$, is (0, 750). This point signifies the value of the function when the input 'x' is zero. In practical terms, for scenarios modeled by this function, the y-intercept often represents the starting point or initial condition.

To calculate the y-intercept, we substitute x = 0 into the function: $y = 750(1.4)^0 = 750 * 1 = 750$. This confirms that the function starts at the point (0, 750) on the coordinate plane.

The x-intercept, on the other hand, represents the point(s) where the function's output 'y' is zero. For $y = 750(1.4)^x$, we established that there is no x-intercept, denoted as DNE. This is because the function's range is restricted to positive values, meaning 'y' will never equal zero.

The absence of an x-intercept is a common characteristic of exponential growth functions. As 'x' decreases towards negative infinity, the function approaches the x-axis but never intersects it. This asymptotic behavior is a defining feature of exponential growth and underscores the function's unbounded nature in the positive y-direction.

In contrast, exponential decay functions, where the base 'b' is between 0 and 1, also typically lack x-intercepts for the same reason: their output 'y' remains positive. However, if the function were vertically translated downwards (e.g., $y = 750(1.4)^x - 1000$), it could potentially have an x-intercept, depending on the magnitude of the translation.

In conclusion, our comprehensive analysis of the exponential function $y = 750(1.4)^x$ has yielded a clear understanding of its key characteristics. By meticulously examining its domain, range, and intercepts, we have constructed a table that succinctly summarizes the function's behavior. The domain, spanning all real numbers $(-\infty, \infty)$, reflects the function's ability to accept any input value for 'x'. The range, confined to positive real numbers $(0, \infty)$, highlights the function's inherent property of exponential growth, where the output 'y' remains strictly positive. The y-intercept, (0, 750), marks the initial value of the function, providing a crucial reference point on the coordinate plane. The absence of an x-intercept, denoted as DNE, underscores the function's asymptotic behavior, approaching but never intersecting the x-axis.

This detailed exploration not only illuminates the specific attributes of $y = 750(1.4)^x$ but also provides a template for analyzing other exponential functions. By understanding the roles of the base 'b' and the coefficient 'a', we can readily determine the domain, range, and intercepts of various exponential functions. Furthermore, the concepts of exponential growth and decay, horizontal asymptotes, and the significance of intercepts are fundamental principles in mathematics and have broad applications in diverse fields.

The completed table serves as a valuable tool for visualizing and interpreting the behavior of the exponential function. It encapsulates the essence of the function in a concise format, facilitating quick reference and deeper understanding. This exercise underscores the importance of analytical skills in mathematics, enabling us to dissect complex functions and extract meaningful information. Ultimately, a thorough grasp of exponential functions is essential for students, researchers, and professionals seeking to model and understand real-world phenomena characterized by rapid growth or decay.