The given function in the image is $f(x) = \sqrt{x}$. According to the problem, the function is to be translated 2 units to the left and 1 unit down.

Step-by-step solution:

1. Horizontal Translation (Left by 2 units): To move a function $f(x)$ 2 units to the left, we replace $x$ with $x + 2$. Therefore, the new function becomes: $f(x) = \sqrt{x + 2}$

2. Vertical Translation (Down by 1 unit): To move a function down by 1 unit, we subtract 1 from the entire function. So, the new function is: $f(x) = \sqrt{x + 2} - 1$

Final Transformed Function:

Thus, the final form of the function after the translation is: $f(x) = \sqrt{x + 2} - 1$

Would you like further details or have any questions?

Related Questions:

1. How would the function change if it were translated 3 units to the right instead of left?
2. What happens to the domain of the function $f(x) = \sqrt{x}$ after translation?
3. Can this transformation be represented graphically? How does the graph look after translation?
4. How do you reflect a square root function across the y-axis?
5. How do you find the inverse of $f(x) = \sqrt{x}$?

Tip:

When translating functions, always adjust the variable inside the function for horizontal shifts and adjust the entire function output for vertical shifts.