A function f is given.
f(x) = 8 − 2x,    1 < x < 3
(a)
Sketch a graph of f.

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.

A function f is given. f(x) = 8 − 2x, 1 < x < 3 (a) Sketch a graph of f.

Learn how to sketch the graph of the linear function f(x) = 8 - 2x for the interval 1 < x < 3. This step-by-step solution demonstrates key concepts such as linear functions, slope, and graphing, and is suitable for students in Grades 9-10.

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