91Ó°ÊÓ

Understanding domain and range is essential when dealing with functions. The **domain** of a function is the set of all possible input values (x-values) that the function can accept. For example, the domain of the linear function \( y_2 = 3x - 4 \) is all real numbers, which we write as \( (-\infty, \infty) \). This is because a linear function, like a straight line, extends indefinitely in both directions.

The **range** is the set of all possible output values (y-values) produced by the function. To find the range, we consider the function's behavior. For the composite function \( y_3 = \sqrt{y_2 + 1} \), first, we note that \( y_3 \) must be non-negative since a square root cannot produce a negative number. Hence, the range for the square root function starts from zero and increases: \( [0, \infty) \).

Understanding domain and range helps in accurately depicting and analyzing functions, whether simple or composite.

A graphing calculator is an invaluable tool for visualizing mathematical functions. By helping you input functions and view their graphs, it gives a real-time look at how the functions behave.

In our exercise, we first define the functions:

• \( y_1 = \sqrt{x + 1} \)
• \( y_2 = 3x - 4 \)

Next, we use the calculator to define the composite function \( y_3 = \sqrt{y_2 + 1} \). When you graph these functions, it becomes easier to understand the relationship between them and visually interpret the domain and range.

Using a graphing calculator simplifies complex calculations and offers a visual representation, aiding in deeper comprehension.

The square root function, represented as \( y = \sqrt{x} \), is a common mathematical function. It only produces non-negative results because it represents the principal (positive) square root.

For instance, in our exercise where \( y_3 = \sqrt{y_2 + 1} \), the function requires that the expression within the square root, \( y_2 + 1 \), be zero or greater. Thus, solving for this gives the domain constraints for \( y_3 \).

To solve:

• Set the expression inside the square root to be non-negative: \( 3x - 3 \geq 0 \)
• This results in \( x \geq 1 \)

Consequently, the domain of \( y_3 \) is \( [1, \infty) \).

The range of the square root function starts from zero and extends to infinity, reflecting that it only produces non-negative outputs: \( [0, \infty) \).

A linear function is represented as \( y = mx + b \), where **m** is the slope and **b** is the y-intercept. Linear functions graph as straight lines and possess a domain and range of all real numbers, given they extend indefinitely in both directions.

For example, consider \( y_2 = 3x - 4 \):

• The slope (**m**) is 3, indicating the line rises 3 units for every unit increase in x.
• The y-intercept (**b**) is -4, meaning the line crosses the y-axis at (0, -4).

Since \( y_2 \) is linear, its domain is \( (-\infty, \infty) \).

When composing functions, understanding linear functions helps precisely predict the behavior of composite functions like \( y_3 = \sqrt{3x - 3} \).