The function f defined by:
[tex]f(x)=-\sqrt{x-b}+a[/tex]
Since the square root of a negative number is undefined, it follows that the function f is defined for all x such that:
[tex]\begin{gathered} x-b\geq0 \\ \text{ Add }b\text{ to both sides of the inequality:} \\ x-b+b\geq0+b \\ x\geq b \end{gathered}[/tex]
Therefore, the domain of the function f is [b, ∞)
The value of f(x) at x=b is given by:
[tex]f(b)=-\sqrt{b-b}+a=a[/tex]
Therefore, the maximum value of f(x)=a and the minimum is infinitely small.
Hence, the range of the function f is (-∞, a].
Therefore, the correct answer is choice B.
