Question

Describe the domain and range. Graph the function. What do I do with the -3? The graph shows that y and x are at 0 so...?

Answer 1

Answers:

Domain: all real numbers

Range: $y \le 6$

Graph is below.

====================================================

Explanation:

This is a piecewise function that is composed of two parts:

1. The first piece being y = x+6 but only when $x \le 0$
2. The second piece occurs when x > 0 and that piece is y = -3x

This piecewise function has split identities or multiple aliases so to speak. It all depends on what the x input is.

The graph is shown below.

The domain is the set of all real numbers since any x value will lead to some y output. Even x = 0 itself is allowed. Note the arrows indicate the graph stretches on forever to the left and to the right.

The range on the other hand isn't all real numbers. Instead, it's $y \le 6$ indicating that y = 6 is the largest y can get. Simply locate the highest point on the graph (similar to the concept of the vertex) and you'll spot the highest y value.

Side notes:

• The domain in interval notation is $(-\infty, \infty)$ which encompasses the entire real number line.
• The range in interval notation is $(-\infty, 6]$ where the square bracket includes the endpoint 6.