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Let's go through each function to determine its domain and range.

### 1. \( f(x) = \sqrt{2x + 1} \)
   - **Domain**: The expression under the square root must be non-negative, so:
     \[
     2x + 1 \geq 0
     \]
     \[
     x \geq -\frac{1}{2}
     \]
     Therefore, the domain is \( x \in \left[ -\frac{1}{2}, \infty \right) \).
   - **Range**: The square root function produces non-negative values. Since \(\sqrt{2x + 1} \geq 0\) for all \(x\) in the domain, and \(2x + 1\) can grow indefinitely, the range is:
     \[
     f(x) \in [0, \infty)
     \]

### 2. \( f(x) = 3x^2 + 1 \)
   - **Domain**: This is a polynomial function, so it is defined for all real values of \(x\):
     \[
     x \in (-\infty, \infty)
     \]
   - **Range**: Since \(3x^2 + 1\) is a quadratic function that opens upwards (coefficient of \(x^2\) is positive), the minimum value occurs at the vertex. The vertex of \(y = 3x^2 + 1\) is at \(x = 0\):
     \[
     f(0) = 3(0)^2 + 1 = 1
     \]
     As \(x\) goes to \(\pm \infty\), \(3x^2 + 1\) increases without bound. Therefore, the range is:
     \[
     f(x) \in [1, \infty)
     \]

### 3. \( f(x) = 2|x - 3| \)
   - **Domain**: The absolute value function is defined for all real \(x\), so:
     \[
     x \in (-\infty, \infty)
     \]
   - **Range**: The function \(2|x - 3|\) is always non-negative since it is a scaled absolute value function. The minimum value occurs when \(x = 3\):
     \[
     f(3) = 2|3 - 3| = 0
     \]
     As \(x\) moves away from 3, \(2|x - 3|\) increases without bound. Thus, the range is:
     \[
     f(x) \in [0, \infty)
     \]

### Summary Table
Here's the completed table for the domain and range of each function:

\[
\begin{array}{|c|c|c|c|}
\hline
 & f(x) = \sqrt{2x + 1} & f(x) = 3x^2 + 1 & f(x) = 2|x - 3| \\
\hline
\text{Domain} & \left[ -\frac{1}{2}, \infty \right) & (-\infty, \infty) & (-\infty, \infty) \\
\hline
\text{Range} & [0, \infty) & [1, \infty) & [0, \infty) \\
\hline
\end{array}
\]

Would you like any additional explanations, or have further questions on any part of this solution?

### Further Questions:
1. How can we determine the range of a function if it has no obvious minimum or maximum point?
2. What is the significance of the domain restrictions for functions involving square roots or absolute values?
3. How do the shapes of quadratic functions affect their range?
4. How would the domain and range change if we modified the functions slightly, for instance, \( f(x) = \sqrt{2x + 5} \)?
5. What are other methods to find the range of functions besides calculating minimum and maximum values?

### Tip:
When working with absolute value functions like \( f(x) = 2|x - 3| \), always remember that they will yield non-negative outputs, making it straightforward to determine the range as starting from zero.

Find the domain and range of each function: f(x) = √(2x + 1), f(x) = 3x^2 + 1, f(x) = 2|x - 3|.

Learn how to find the domain and range of functions like f(x) = √(2x + 1), f(x) = 3x^2 + 1, and f(x) = 2|x - 3|. This detailed solution covers key concepts such as square roots, quadratic functions, and absolute value properties. Suitable for Grades 9-11 students.

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