Let's first break down the function $f(x) = 8 - 2x$ defined for $1 < x < 3$.

Steps to sketch the graph:

1. Identify the type of function:

   • This is a linear function since it has the form $y = mx + b$, where $m = -2$ and $b = 8$.
   • The slope $m = -2$ means the line is decreasing, and the $y$-intercept is $8$ (although we are only concerned with the interval $1 < x < 3$).

2. Determine the values of the function at key points:

   • At $x = 1$: $f(1) = 8 - 2(1) = 6$
   • At $x = 3$: $f(3) = 8 - 2(3) = 2$

3. Plot the points:

   • $(1, 6)$ and $(3, 2)$.

4. Draw the line segment:

   • Since the domain is restricted to $1 < x < 3$, the graph will be a straight line segment between $(1, 6)$ and $(3, 2)$.

Graph properties:

• Slope: $-2$ (the line will descend from left to right).
• Y-values range from 6 to 2.
• The graph is only valid for $1 < x < 3$, so the line should not extend beyond this interval.

I can visualize this for you by generating a graph plot if you'd like! Would you like to see it?

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Here are 5 related questions:

1. What happens to the graph if the slope is increased to $-1$?
2. How would the graph change if the domain was $0 < x < 4$?
3. What is the equation of the line perpendicular to $f(x)$ passing through $(2, f(2))$?
4. How can you find the inverse of $f(x)$?
5. What are the intercepts of $f(x)$ outside of the given domain?

Tip: When sketching linear functions, always check the slope and intercept, and focus on key points in the defined domain.