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Let's explore the concept of the domain of a function first. The domain refers to all the possible input values (usually represented by x) that can be plugged into a function. For a linear function like \( f(x) = 9x - 2 \), the domain is quite simple. There are no restrictions on the values that x can take because we don't run into issues like dividing by zero or taking the square root of a negative number. Thus, the domain of \( f(x) = 9x - 2 \) is all real numbers, which we write as \( (-\infty, \infty) \). In essence, you can choose any real number to substitute for x, and the function f will give you a valid output.

Next, let's delve into the idea of the range of a function. The range includes all the possible output values (usually represented by f(x) or y) that the function can produce. For our linear function \( f(x) = 9x - 2 \), it is easier to see by considering its graph. A linear function like this one forms a straight line that extends infinitely in both upward and downward directions on the graph. Because of this characteristic, our function can take on any real number as an output. This means that the range of \( f(x) = 9x - 2 \) is all real numbers, denoted by \( (-\infty, \infty) \). Anytime you encounter a linear function, you can be sure that both the domain and the range include all real numbers.

Finally, let's clarify what we mean by real numbers. In mathematics, real numbers encompass a vast set of numbers that include both rational (like 3, -1/2, 4.5) and irrational numbers (like \( \sqrt{2} \), \( \pi \)). They can be either positive, negative, or zero. The real numbers make up the entire number line, meaning any point you can find on the number line represents a real number. This is why real numbers are so significant in understanding the domain and range of functions; they cover all conceivable values you might plug in or get out of a function, like our example \( f(x) = 9x - 2 \). Working with real numbers allows us to make sense of continuous data and to graph functions accurately.