We are asked to graph the composite function $g(f(x))$, where:

$f(x) = -2x - 5 \quad \text{and} \quad g(x) = x - 1$

Step 1: Find the composite function $g(f(x))$

To compute $g(f(x))$, we substitute $f(x)$ into $g(x)$. Specifically, since $g(x) = x - 1$, we replace $x$ in $g(x)$ with $f(x)$:

$g(f(x)) = g(-2x - 5) = (-2x - 5) - 1$

Simplifying this:

$g(f(x)) = -2x - 5 - 1 = -2x - 6$

Step 2: Graphing the function

The composite function $g(f(x)) = -2x - 6$ is a linear function. Its slope is $-2$, and its y-intercept is $-6$.

To graph this, we follow these steps:

1. Plot the y-intercept at $(0, -6)$.
2. Use the slope $-2$, which indicates that for every 1 unit increase in $x$, $y$ decreases by 2 units. From the y-intercept, move 1 unit to the right and 2 units down to find another point on the line.
3. Connect these points to form a straight line.

Would you like me to generate a graph for this function?

Follow-up questions:

1. How do you compute the composite of two functions in general?
2. What is the significance of the slope in a linear function?
3. How does the y-intercept influence the graph of a linear equation?
4. What is the difference between $f(g(x))$ and $g(f(x))$?
5. Can composite functions always be simplified into basic forms like linear functions?

Tip: Always substitute carefully when computing composite functions to avoid errors in simplification!