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 \