Question

the function f is defined by f(x)=√x. write down the expression for k(x).

Explanation:

Step1: Analyze the graph transformation

The parent function is \( f(x)=\sqrt{x} \). The graph of \( k(x) \) seems to be a transformation of \( f(x) \). Looking at the graph, the vertex of \( k(x) \) is at \( (4, -2) \), and it appears to be a horizontal shift and vertical shift of \( f(x) \). Also, the shape suggests a horizontal shift right by 4 units and vertical shift down by 2 units, or maybe a transformation related to the argument and the function value. Wait, another way: let's check the domain and the transformation. The graph of \( k(x) \) starts at \( x = 4 \) (since the point is at \( (4, -2) \)) and then increases, similar to \( \sqrt{x} \) but shifted. So if we consider the transformation, for a square root function, a horizontal shift right by \( h \) units and vertical shift down by \( k \) units would be \( y=\sqrt{x - h}-k \). From the graph, the starting point (vertex) is at \( (4, -2) \), so \( h = 4 \), \( k = 2 \). So the function \( k(x)=\sqrt{x - 4}-2 \). Let's verify: when \( x = 4 \), \( k(4)=\sqrt{4 - 4}-2=0 - 2=-2 \), which matches the point. When \( x = 8 \), \( k(8)=\sqrt{8 - 4}-2=\sqrt{4}-2 = 2 - 2 = 0 \), which also matches the graph (the point at \( (8,0) \)). So the transformation is a horizontal shift right 4 and vertical shift down 2.

Step2: Write the expression

So combining the transformations, the function \( k(x) \) is \( \sqrt{x - 4}-2 \).

Answer:

\( \sqrt{x - 4}-2 \)