transformation, domain and range of f(x) = |x| +2

### Transformation:
The function \( f(x) = |x| + 2 \) is a transformation of the basic absolute value function \( y = |x| \). The transformations involved are:

1. **Vertical Shift:** The graph of \( y = |x| \) is shifted **upward** by 2 units. This shift is due to the "+2" added to the function, which moves every point on the graph vertically by 2 units.

The resulting graph retains the "V" shape of the absolute value function but is now positioned higher on the y-axis.

### Domain:
The domain of \( f(x) = |x| + 2 \) remains the same as that of the basic absolute value function \( y = |x| \) because the absolute value function is defined for all real numbers:
\[
\text{Domain: } (-\infty, \infty)
\]

### Range:
The range of \( f(x) = |x| + 2 \) is affected by the vertical shift. Since the original minimum value of \( y = |x| \) is 0, after shifting the graph up by 2 units, the new minimum value is 2. Thus, the range of the function is all real numbers greater than or equal to 2:
\[
\text{Range: } [2, \infty)
\]

### Summary:
- **Transformation:** Vertical shift upward by 2 units.
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( [2, \infty) \)

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the transformation change for \( f(x) = |x| - 3 \)?
2. What is the effect of adding a constant inside the absolute value, such as in \( f(x) = |x + 1| + 2 \)?
3. How do horizontal shifts affect the domain and range of absolute value functions?
4. What would be the vertex of the function \( f(x) = |x - 4| + 2 \)?
5. Can you describe the graph of \( f(x) = -|x| + 2 \)?

**Tip:** When analyzing transformations of functions, pay close attention to operations outside the function (e.g., "+2" for vertical shifts) and inside the function (e.g., "|x + 1|") to understand their impact on the graph.

Explore the transformation, domain, and range of the function f(x) = |x| + 2. Learn about vertical shifts in absolute value functions and how they affect the graph. Suitable for high school students and anyone interested in mathematical functions.

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