Let's break down the two functions you've provided:

1. $f(x) = -3x - 8$:

   • This is a linear function. Its graph is a straight line with slope $-3$ and y-intercept $-8$.
   • The slope of $-3$ means the line will decrease by 3 units in the y-direction for every 1 unit increase in the x-direction.
   • The y-intercept $-8$ means that the line crosses the y-axis at $(0, -8)$.

2. $g(x) = \sqrt{-x}$:

   • This is a square root function with a domain restriction. Since you cannot take the square root of a negative number (in the real number system), the domain of this function is limited to $x \leq 0$. In other words, $g(x)$ is only defined for negative values of $x$ and at $x = 0$.
   • The graph of this function will start at the point $(0, 0)$ and move leftward, decreasing as $x$ gets more negative.

Key Observations:

• Domain of $g(x)$: $(-\infty, 0]$, because for any positive $x$, $g(x)$ is not defined.
• Range of $g(x)$: $[0, \infty)$, because the square root of any negative number results in a non-negative value.
• Graph of $f(x)$: A straight line with slope $-3$ and y-intercept $-8$.
• Graph of $g(x)$: A curve that starts at $(0, 0)$ and decreases as $x$ moves leftward.

Would you like to explore how these functions interact or perform any specific operations like finding their intersection points?

Expanding Questions:

1. How do you find the intersection points of $f(x)$ and $g(x)$?
2. How do you calculate the domain and range of other types of functions?
3. What happens to the graph of $g(x) = \sqrt{-x}$ if we replace $-x$ with $x$?
4. How do transformations affect the graph of $f(x) = -3x - 8$ if we add or subtract constants?
5. How do you find the inverse of linear functions like $f(x)$?

Tip:

To graph functions like $g(x) = \sqrt{-x}$, it's useful to first identify its domain and range to understand the behavior of the graph and restrict the x-values accordingly.