The problem states that the function $f(x) = \sqrt{x}$ is translated 2 units to the left and 1 unit downward. We need to find the new form of the function after the translation.

Translation steps:

1. Horizontal translation: Moving the function 2 units to the left involves replacing $x$ with $x + 2$.

   • So, $f(x) = \sqrt{x}$ becomes $f(x) = \sqrt{x + 2}$.

2. Vertical translation: Moving the function 1 unit downward involves subtracting 1 from the function.

   • Now, the new function becomes $f(x) = \sqrt{x + 2} - 1$.

Final Result:

The translated function is: $f(x) = \sqrt{x + 2} - 1$

Would you like further explanation, or do you have any other questions?

Related questions:

1. What happens if we translate the function 3 units to the right instead?
2. How does a vertical translation affect the domain of a square root function?
3. Can you explain the effect of translating a quadratic function instead of a square root function?
4. What is the graph of the function $f(x) = \sqrt{x + 2} - 1$ like?
5. How do horizontal and vertical translations impact the range of a function?

Tip:

When translating a function, always replace $x$ inside the function for horizontal shifts and adjust the whole function for vertical shifts!