Question

Find (a) The domain. (b) The range. $$ f(x)=x-3 $$

Short answer

Answer: The domain and the range of the function \(f(x) = x - 3\) are both all real numbers, represented as \((-\infty, \infty)\).

Step-by-step solution

Find the domain of the function

To find the domain of the function \(f(x) = x - 3\), we must determine the set of all possible values of \(x\) for which the function is defined. In this case, there is no division by zero or square roots of a negative number involved, so the function is defined for all real numbers. Therefore, the domain of the function is all real numbers, which can be represented as \((-\infty, \infty)\).

Find the range of the function

To find the range of the function \(f(x) = x - 3\), we must determine the set of all possible values of \(f(x)\) (or y-values) that can result from using the x-values of the domain. Since this function is a linear function with a slope of 1 and no restrictions on the input, it will produce all possible y-values as well. Therefore, the range of the function is also all real numbers, which can be represented as \((-\infty, \infty)\). In conclusion, the domain and the range of the function \(f(x) = x - 3\) are both all real numbers, represented as \((-\infty, \infty)\).

Key concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (or x-values) that the function can accept. In simpler terms, it’s about finding the values for which our function is defined without any issues. For the linear function \(f(x) = x - 3\), we check for common restrictions like division by zero or square roots with negative numbers, but neither exist here. Thus, there are no restrictions on \(x\), and the function is defined for every possible x-value. Consequently, the domain of this linear function is all real numbers, which we can write in interval notation as \((-fty, ft)\). Understanding the domain is important as it tells us the full scope of inputs the function can handle.

Range of a Function

The range of a function is just as important as the domain. It refers to all possible output values (or y-values) that result when we input domain values into the function. For the linear function \(f(x) = x - 3\), we need to examine the y-values that can be achieved. Because this is a simple linear equation with a slope of 1 and no bounds on \(x\), every real number can be produced as an output. Therefore, much like the domain, the range for this function is also all real numbers. This is again written as \((-fty, ft)\). Recognizing the range is essential as it defines the spread of y-values our function is capable of generating.

Real Numbers

In mathematics, real numbers encompass a vast set of numbers that include both rational and irrational numbers. Key characteristics of real numbers include:

• They can be positive, negative, or zero.
• They include fractions and decimals.
• They don’t include imaginary numbers.

Real numbers are pivotal in functions like \(f(x) = x - 3\), as they form the basis for defining both domain and range. Every input (or \(x\)) and output (or \(f(x)\)) in this function is a real number, explaining why the domain and range span all real numbers \((-fty, fty)\). Understanding real numbers and their role in mathematical functions helps in grasping broader concepts like algebra and calculus.