graph f(x)=square root x-3 and identify its key functions

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.

Learn how to graph the function f(x) = √(x - 3) and identify its key features such as domain, range, intercepts, and graph shape. This detailed solution explores the behavior of the square root function and its transformations. Suitable for high school students in Grades 9-11.

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