Technology
Exploring the Domain and Range of f(x) e^(x-1) - 1
Exploring the Domain and Range of f(x) e^(x-1) - 1
Introduction
Understanding the domain and range of a function is fundamental in mathematics, providing insights into the behavior and limitations of the function. In this article, we will delve into the domain and range of the function f(x) ex-1 - 1. We will explore the mathematical properties of this exponential function and determine its domain and range.
Defining the Function
The function in question is f(x) ex-1 - 1, where e is the base of the natural logarithm, approximately equal to 2.71828. This function involves an exponential component, making it a critical case for understanding domain and range in the context of exponential functions.
Finding the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) ex-1 - 1, the exponential function ex-1 is defined for all real numbers. Therefore, the domain of f(x) is all real numbers, denoted as R or (-∞, ∞).
Finding the Range
The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range of f(x) ex-1 - 1, we need to analyze the behavior of the function as the input x varies over its domain.
Using the Inverse Function
To find the range, we can use the concept of the inverse function. If we can find the inverse of f(x), we can determine its range by finding the domain of the inverse function.
Starting with the equation:
y ex-1 - 1
We solve for x in terms of y to find the inverse function:
y 1 ex-1
ln(y 1) x - 1
x ln(y 1) 1
Therefore, the inverse function is:
f-1(x) ln(x 1) 1
The domain of the inverse function f-1(x) is the set of all x such that x 1 > 0, which simplifies to x > -1. Hence, the range of the original function f(x) ex-1 - 1 is (-1, ∞).
Mathematical Verification
To verify that the range is (-1, ∞), we can check the limits of the function as x approaches positive and negative infinity:
As x approaches ∞:
limx→∞ f(x) limx→∞ ex-1 - 1 ∞
As x approaches -∞:
limx→-∞ f(x) ex-1 - 1 -1
Thus, the range of f(x) ex-1 - 1 is indeed (-1, ∞).
Summary
In conclusion, the function f(x) ex-1 - 1 has a domain of all real numbers R and a range of (-1, ∞). This result is verified by both the concept of the inverse function and the evaluation of the limits of the function. Understanding the domain and range of this function is crucial for further mathematical analysis and applications.