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The image contains a mathematical function \( f(x) = 3 \sqrt{-2x + 1} \), along with sections to analyze its domain, range, extrema, increasing and decreasing intervals, and end behavior. It also asks for parent function identification and transformations.

Let's go step by step:

### 1. **Parent Function:**
The given function is a transformation of the square root function \( \sqrt{x} \). So, the parent function is:
\[
f_{\text{parent}}(x) = \sqrt{x}
\]

### 2. **Transformations:**
The given function is \( f(x) = 3 \sqrt{-2x + 1} \). We can break down the transformations applied to the parent function \( \sqrt{x} \):
- **Horizontal stretch/compression and reflection**: The term \( -2x \) indicates a horizontal reflection across the y-axis and a horizontal compression by a factor of 2.
- **Translation**: The \( +1 \) inside the square root shifts the graph horizontally.
- **Vertical stretch**: The coefficient 3 outside the square root indicates a vertical stretch by a factor of 3.

### 3. **Domain:**
To find the domain, we need to determine where the expression inside the square root is non-negative (since the square root of a negative number is not real):
\[
-2x + 1 \geq 0
\]
Solving for \( x \):
\[
-2x \geq -1 \quad \Rightarrow \quad x \leq \frac{1}{2}
\]
Thus, the domain is:
\[
(-\infty, \frac{1}{2}]
\]

### 4. **Range:**
Since the function has a vertical stretch and starts from a minimum point (shifted vertically), the range is all values starting from 0 (the square root of 0) and extending upward, multiplied

Analyze the function f(x) = 3√(-2x + 1) by finding its parent function, transformations, domain, range, extrema, increasing and decreasing intervals, and end behavior.

Explore the function f(x) = 3√(-2x + 1) with a detailed analysis of its parent function, transformations, domain, range, increasing and decreasing intervals, and end behavior. This problem is suitable for high school students in Grades 10-12.

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