Question

Find (a) The domain. (b) The range. $$ y=7 $$

Short answer

Answer: The domain of the horizontal line y=7 is all real numbers, represented as (-∞, ∞), and the range is the single value 7, represented as {7}.

Step-by-step solution

Identify the function as a horizontal line

The equation given is y=7, which is a horizontal line. This means that the line is flat and parallel to the x-axis.

Determine the domain of the function

The domain of y=7 consists of all real numbers for x. Since the line is horizontal, it extends infinitely in both left and right directions along the x-axis. This means that the domain is: $$ D = (-\infty, \infty) $$

Determine the range of the function

The range of y=7 is just the single value of y, which is equal to 7. Since the line is flat and parallel to the x-axis, its y-value remains constant. This means that the range is: $$ R = \{7\} $$ In conclusion, for the horizontal line y=7, the domain is \(D=(-\infty, \infty)\) and the range is \(R=\{7\}\).

Key concepts

Domain of a Function

When we talk about the domain of a function, we're discussing the set of all possible input values (x-values) that the function can accept. For a horizontal line, like the function given by the equation \(y=7\), the domain is quite straightforward.
Essentially, this line extends in both directions along the x-axis endlessly since it's flat and parallel to the x-axis.

• This means it covers every possible x-value you can think of.
• There are no restrictions or exclusions for x.

Therefore, the domain of the function \(y=7\) is all real numbers, expressed mathematically as \(D=(-\infty, \infty)\). This notation indicates that as x moves in the positive and negative directions without bound, it can take any real number as its value.

Range of a Function

Now let's explore the range of a function, which refers to all possible output values (y-values) that a function can produce. In the case of the horizontal line \(y=7\), the range is actually a single element.
Because it's a horizontal line, its y-value doesn’t change no matter what x-value you choose.

• Every point on this line has the same y-coordinate, which is 7.
• Thus, there is no variation or spread in the y-values of the function.

Given this, the range of our function \(y=7\) is simply \(R=\{7\}\). This indicates that the only output value the function can produce is y=7, regardless of what x is.

Real Numbers

Real numbers are quite crucial when we're discussing functions, domains, and ranges. But what exactly are real numbers? Essentially, they include every type of number you can think of except for imaginary numbers.
This encompasses a very broad set of numbers:

• All whole numbers like 1, 2, 3, and so on.
• Fractions and decimals, such as 1/2 or 3.14.
• Even negative numbers, like -5 or -0.75.

Real numbers are necessary for defining the domain and range of functions. When we say the domain of a function is all real numbers, it means x can be anything within this expansive set. Understanding real numbers helps you grasp the wide scope of values x can assume, especially in cases involving horizontal and vertical lines where there are fewer restrictions on values.