To graph the function $f(x) = \sqrt{x - 3}$, let's first understand its structure and key features:

Key Features:

1. Domain:

   • The square root function is only defined for non-negative values. For $f(x) = \sqrt{x - 3}$, the expression inside the square root, $x - 3$, must be greater than or equal to 0.
   • Hence, $x - 3 \geq 0$, so $x \geq 3$.
   • Therefore, the domain of the function is $x \geq 3$ or $[3, \infty)$.

2. Range:

   • The square root function outputs only non-negative values (since the square root of any real number is non-negative).
   • Therefore, the range of $f(x)$ is $[0, \infty)$.

3. Intercepts:

   • x-intercept: The function will cross the x-axis when $f(x) = 0$, so we set $\sqrt{x - 3} = 0$, which gives $x - 3 = 0$, or $x = 3$. Hence, the x-intercept is at $(3, 0)$.
   • y-intercept: Since the domain starts at $x = 3$, there is no y-intercept.

4. Shape of the graph:

   • The function starts at $(3, 0)$ and increases gradually as $x$ increases. The graph will have a slow, upward curve that flattens out as $x$ gets larger.

Graph Sketch:

• The graph begins at $x = 3$, where $f(3) = 0$, and as $x$ increases, $f(x)$ increases, but at a decreasing rate.

Let me now graph the function for you.

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Would you like a visual of the graph, or would you prefer to dive deeper into any specific aspects of this function? Here are some related questions that might interest you:

1. How do transformations like shifts affect the square root function?
2. How would the graph change if the function was $f(x) = \sqrt{x + 3}$?
3. How can we algebraically solve for the inverse of $f(x) = \sqrt{x - 3}$?
4. What happens if the function has a negative sign in front, like $f(x) = -\sqrt{x - 3}$?
5. How does the rate of change of the square root function compare to that of a linear function?

Tip: When graphing functions like square roots, focus on identifying key points like intercepts and the behavior of the function at the endpoints of the domain.