The function \( f(x) = |x| + 3 \) is defined for all real numbers \( x \). Here's a breakdown of what this function represents:

1. **Absolute Value:** The absolute value function \( |x| \) outputs the non-negative value of \( x \). This means:
   - If \( x \geq 0 \), then \( |x| = x \).
   - If \( x < 0 \), then \( |x| = -x \).

2. **Vertical Shift:** The addition of 3 to \( |x| \) shifts the entire graph of \( |x| \) upward by 3 units.

### Graph of the Function
The graph of \( f(x) = |x| + 3 \) is a V-shaped curve that:
- **Vertex:** Has a vertex at \( (0, 3) \).
- **Slope:** For \( x \geq 0 \), the slope of the graph is 1 (increasing linearly).
- **Slope:** For \( x < 0 \), the slope of the graph is -1 (decreasing linearly).

### Key Points
- \( f(0) = 3 \)
- \( f(1) = 4 \)
- \( f(-1) = 4 \)
- \( f(2) = 5 \)
- \( f(-2) = 5 \)

### Domain and Range
- **Domain:** \( (-\infty, \infty) \) (all real numbers)
- **Range:** \( [3, \infty) \) (all real numbers greater than or equal to 3)

Would you like more details or have any specific questions?

Here are 5 related questions to expand on this concept:
1. What would the graph of \( f(x) = |x| - 3 \) look like?
2. How would you find the points where \( f(x) = 5 \)?
3. What is the derivative of \( f(x) = |x| + 3 \), and how does it change at \( x = 0 \)?
4. Can you express \( f(x) = |x| + 3 \) as a piecewise function?
5. How would the graph change if the function was \( f(x) = 2|x| + 3 \)?

**Tip:** The absolute value function is useful for representing situations where only the magnitude of a quantity matters, regardless of direction.

Explore the function f(x) = |x| + 3 in detail, including its graph, key points, domain, and range. Learn how the absolute value function and vertical shifts affect the graph and behavior of the function. Suitable for high school students.

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