The problem involves a transformation of the function $f(x) = \sqrt{x}$. You are asked to determine the equation of the function $k(x)$, which is a translation of $f(x)$.

From the image, we can observe that the graph of $f(x)$ is shifted down and to the right, meaning it has undergone a horizontal and vertical translation.

Let's break this down:

• The function $f(x) = \sqrt{x}$ starts at the origin (0, 0).

• The graph of $k(x)$, however, appears to be shifted both horizontally and vertically.

  From the graph:

  • The graph of $f(x)$ is shifted 4 units to the right (positive horizontal shift).
  • It is also shifted 3 units down (negative vertical shift).

Using the general form for transformations:

$k(x) = f(x - h) + k$

where $h$ is the horizontal shift and $k$ is the vertical shift, we have:

$k(x) = \sqrt{x - 4} - 3$

This reflects the translation of the square root function by 4 units to the right and 3 units down.

So, the expression for $k(x)$ is:

$k(x) = \sqrt{x - 4} - 3$

Would you like further clarification on this transformation?

Related Questions:

1. How would the function change if it was translated up instead of down?
2. What happens to the domain of $f(x)$ after this translation?
3. How would you write the function if it were shifted 5 units to the left and 2 units up?
4. Can you identify the vertex of the translated function $k(x)$?
5. How does the horizontal translation affect the square root function's range?

Tip: Horizontal shifts inside the function affect the input, while vertical shifts outside the function affect the output directly!