Question

3
select the correct answer.
the graph shows function j, a transformation of ( f(x) = x^{\frac{1}{2}} ).
graph of a curve starting from the left, decreasing, passing through (0, 2), and ending at (4, 0) on the x - axis
which equation represents the graph of function j?
a. ( j(x) = (x - 4)^{\frac{1}{2}} )
b. ( j(x) = -(x + 4)^{\frac{1}{2}} )
c. ( j(x) = (-x + 4)^{\frac{1}{2}} )
d. ( j(x) = (-x - 4)^{\frac{1}{2}} )

Explanation:

Step1: Analyze the original function

The original function is \( f(x) = x^{\frac{1}{2}}=\sqrt{x} \), which has a domain \( x\geq0 \) and is increasing. The graph of \( j(x) \) has a domain (from the graph, the right - most point is at \( x = 4 \)) and the function is decreasing, and when \( x = 0 \), \( j(0)=2 \)? Wait, no, let's check the transformation. First, let's recall the transformations of square - root functions. The square - root function \( y=\sqrt{x} \) has a domain \( x\geq0 \) and starts at \( (0,0) \), increasing.

Step2: Analyze the domain and direction of the graph of \( j(x) \)

The graph of \( j(x) \) has a domain (the right - most \( x \)-value) \( x\leq4 \) (since the graph ends at \( x = 4 \) on the right) and it is decreasing. Let's rewrite the options in terms of the square - root function.

For a square - root function \( y=\sqrt{ax + b} \), the domain is \( ax + b\geq0 \).

Option A: \( j(x)=(x - 4)^{\frac{1}{2}}=\sqrt{x - 4} \), domain \( x-4\geq0\Rightarrow x\geq4 \), and it is increasing (since the coefficient of \( x \) inside the square root is positive), which does not match the graph (the graph has \( x\leq4 \) and is decreasing).

Option B: \( j(x)=-(x + 4)^{\frac{1}{2}}=-\sqrt{x + 4} \), domain \( x + 4\geq0\Rightarrow x\geq - 4 \), and it is decreasing (because of the negative sign), but the domain is \( x\geq - 4 \), which does not match the graph (the graph ends at \( x = 4 \) and is defined for \( x\leq4 \)).

Option C: \( j(x)=(-x + 4)^{\frac{1}{2}}=\sqrt{-x + 4}=\sqrt{-(x - 4)} \), domain \( -x+4\geq0\Rightarrow -x\geq - 4\Rightarrow x\leq4 \), and the function \( y = \sqrt{-x + 4} \) is decreasing (because the derivative \( y^\prime=\frac{-1}{2\sqrt{-x + 4}}<0 \) for \( x<4 \)), which matches the domain (\( x\leq4 \)) and the decreasing nature of the graph.

Option D: \( j(x)=(-x - 4)^{\frac{1}{2}}=\sqrt{-x - 4} \), domain \( -x-4\geq0\Rightarrow -x\geq4\Rightarrow x\leq - 4 \), which does not match the graph (the graph is defined for \( -6\leq x\leq4 \), not just \( x\leq - 4 \)).

Answer:

C. \( j(x)=(-x + 4)^{\frac{1}{2}} \)