https://file.aiquickdraw.com/mathbot/file/17255997273057uykmiwp.jpg

The function given in the image is:

\[
f(x) = 3\sqrt{2x} - 1
\]

We can proceed by analyzing the function step by step:

### 1. **Parent Function:**
The parent function of \( f(x) = 3\sqrt{2x} - 1 \) is \( \sqrt{x} \). This is the basic square root function.

### 2. **Transformations:**
Let's break down the transformations applied to the parent function \( \sqrt{x} \):
- **Horizontal Compression by a factor of \(\frac{1}{2}\):** The factor \(2\) inside the square root affects the \(x\)-axis. It compresses the graph horizontally.
- **Vertical Stretch by a factor of 3:** The coefficient 3 outside the square root stretches the graph vertically.
- **Vertical Shift Down by 1 unit:** The subtraction of 1 shifts the graph downward by 1 unit.

### 3. **Intercepts:**
- **x-intercept:** To find the x-intercept, set \( f(x) = 0 \):
  \[
  3\sqrt{2x} - 1 = 0 \quad \Rightarrow \quad 3\sqrt{2x} = 1 \quad \Rightarrow \quad \sqrt{2x} = \frac{1}{3} \quad \Rightarrow \quad 2x = \frac{1}{9} \quad \Rightarrow \quad x = \frac{1}{18}
  \]
  So, the x-intercept is \( \left( \frac{1}{18}, 0 \right) \).
  
- **y-intercept:** Set \( x = 0 \):
  \[
  f(0) = 3\sqrt{0} - 1 = -1
  \]
  So, the y-intercept is \( (0, -1) \).

### 4. **Domain:**
The domain is determined by the square root function, where the expression inside the square root must be non-negative:
\[
2x \geq 0 \quad \Rightarrow \quad x \geq 0
\]
Thus, the domain is \( [0, \infty) \).

### 5. **Range:**
The range is based on the transformation:
- The square root function \( \sqrt{x} \) has a range of \( [0, \infty) \).
- The vertical transformation \( 3\sqrt{2x} - 1 \) shifts the graph down by 1, so the minimum value is \( -1 \).
Thus, the range is \( [-1, \infty) \).

### 6. **End Behavior:**
- As \( x \to \infty \), \( f(x) \to \infty \

Analyze the function f(x) = 3√(2x) - 1 to determine its parent function, transformations, intercepts, domain, range, end behavior, and intervals of increase/decrease.

Learn how to analyze the function f(x) = 3√(2x) - 1, including its parent function, transformations, intercepts, domain, range, end behavior, and intervals of increase/decrease. Suitable for high school students in Grades 10-12.

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