The problem you're working on involves identifying the function $g(x)$ based on the translation of $f(x) = \sqrt{x}$.

The graph of $f(x) = \sqrt{x}$ is shown in red, and the graph of $g(x)$ is shown in green. To find $g(x)$, we need to apply the translation rule between the red graph and the green graph.

Let's examine what type of transformation is applied to the red graph:

• Horizontal shifts: If the graph is shifted left or right, the transformation affects the $x$-coordinate inside the square root.
• Vertical shifts: If the graph is shifted up or down, the transformation affects the entire function by adding or subtracting a constant.

From the image, it appears that the green graph is a horizontal translation of the red graph. Specifically, it looks like the graph is shifted to the right by 4 units.

Thus, the transformation would be: $g(x) = \sqrt{x - 4}$

This is because shifting $f(x) = \sqrt{x}$ 4 units to the right results in $g(x) = \sqrt{x - 4}$.

Answer:

$g(x) = \sqrt{x - 4}$

Would you like more details on function transformations, or do you have any further questions?

Here are 5 related questions you might want to explore:

1. What happens to the graph of $f(x) = \sqrt{x}$ if we shift it to the left by 2 units?
2. How would the graph change if we vertically translate $f(x) = \sqrt{x}$ upwards by 3 units?
3. What is the effect of reflecting $f(x) = \sqrt{x}$ across the x-axis on the equation of the function?
4. How does a vertical stretch affect the graph of $f(x) = \sqrt{x}$?
5. What would be the graph of $g(x) = \sqrt{x + 6}$?

Tip: Horizontal translations affect the input (inside the function), while vertical translations affect the output (outside the function).