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.