what is the mapping rule for g(x)= -1/ (2(x-2))

The function you're working with is \( g(x) = -\frac{1}{2(x - 2)} \). 

To determine the mapping rule, we can break down how the transformations are applied to the base function \( f(x) = \frac{1}{x} \), which is the basic reciprocal function.

### Steps to derive the mapping rule:
1. **Horizontal shift**: The expression \( (x - 2) \) inside the denominator represents a horizontal shift to the right by 2 units. This shifts the graph of \( f(x) = \frac{1}{x} \) two units to the right.
   
2. **Vertical stretch/compression**: The factor \( \frac{1}{2} \) outside the denominator causes a vertical compression by a factor of 2. It scales the function vertically by making it "flatter."

3. **Reflection**: The negative sign \( -\frac{1}{2} \) reflects the function over the x-axis. This flips the graph vertically.

### Final Mapping Rule:
- Start with any point \( (x, y) \) on the graph of \( f(x) = \frac{1}{x} \).
- Shift the x-coordinate 2 units to the right: \( x \mapsto x + 2 \).
- Multiply the y-coordinate by \( -\frac{1}{2} \): \( y \mapsto -\frac{y}{2} \).

Thus, the point \( (x, y) \) on the graph of \( f(x) = \frac{1}{x} \) is mapped to the point \( (x + 2, -\frac{y}{2}) \) on the graph of \( g(x) = -\frac{1}{2(x - 2)} \).

### Summary:
- Horizontal shift right by 2 units.
- Vertical compression by a factor of 2.
- Reflection across the x-axis.

Would you like more details or have any questions?

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Here are 5 questions to deepen your understanding:
1. How would the mapping rule change if the function was \( g(x) = \frac{1}{2(x + 2)} \)?
2. What is the effect of removing the negative sign in \( g(x) = \frac{1}{2(x - 2)} \)?
3. How would the graph change if the coefficient were \( -\frac{1}{3} \) instead of \( -\frac{1}{2} \)?
4. What are the domain and range of the function \( g(x) = -\frac{1}{2(x - 2)} \)?
5. Can you identify the vertical and horizontal asymptotes for this function?

**Tip**: When analyzing transformations, always handle horizontal and vertical shifts before dealing with stretching/compression and reflections.

Understand the mapping rule for the function g(x) = -1/(2(x - 2)). This solution breaks down key transformations, including a horizontal shift by 2 units, vertical compression by a factor of 2, and reflection across the x-axis. Suitable for high school students studying functions and transformations.

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