What is the domain of the following function?
Understanding the domain of a function is crucial in mathematics, as it defines the set of all possible input values for which the function is defined. In this article, we will explore the concept of domain and discuss how to determine it for a given function. By the end, you will have a clearer understanding of what constitutes the domain of a function and how to identify it.
Before diving into the specifics, let’s first define what a function is. A function is a relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output. The domain of a function refers to the set of all possible inputs, while the range is the set of all possible outputs.
To determine the domain of a function, we need to consider various factors, such as the nature of the function, any restrictions on the input values, and the context in which the function is used. Let’s examine some common types of functions and how to find their domains.
For polynomial functions, such as f(x) = x^2 + 2x + 1, the domain is typically all real numbers, denoted as (-∞, ∞). This is because there are no restrictions on the input values, and the function is defined for all real numbers.
In the case of rational functions, such as f(x) = (x^2 + 3x + 2) / (x – 1), the domain is all real numbers except for the value that makes the denominator equal to zero. To find this value, we set the denominator equal to zero and solve for x: x – 1 = 0, which gives us x = 1. Therefore, the domain of this function is (-∞, 1) ∪ (1, ∞).
For trigonometric functions, such as f(x) = sin(x), the domain is also all real numbers. However, some trigonometric functions, like tan(x), have restrictions on their domains due to the presence of vertical asymptotes. For example, the domain of tan(x) is all real numbers except for odd multiples of π/2, such as π/2, 3π/2, 5π/2, and so on.
When dealing with exponential functions, such as f(x) = e^x, the domain is again all real numbers. This is because the exponential function is defined for all real numbers, and there are no restrictions on the input values.
Logarithmic functions, such as f(x) = log(x), have a more specific domain. The domain of a logarithmic function is all positive real numbers, as the logarithm is only defined for positive values. For example, the domain of f(x) = log(x + 2) is all real numbers greater than -2.
In conclusion, determining the domain of a function involves considering the nature of the function, any restrictions on the input values, and the context in which the function is used. By understanding these factors, you can identify the domain of a function and ensure that you are working with a well-defined mathematical object. Whether you are dealing with polynomial, rational, trigonometric, exponential, or logarithmic functions, the key to finding the domain lies in recognizing the underlying principles that govern each type of function.
