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.