Hello!
[tex]\large\boxed{\text{B. } k(x) = (-x + 3)^{\frac{1}{2} }}}[/tex]
We can look at each answer choice to verify whether it has a restricted domain or not:
A. [tex]f(x) = (3x)^{\frac{1}{5}} - 4[/tex]
This is a function similar to a cubic root function. Its domain is (-∞, ∞), so it has no restriction.
B. [tex]k(x) = (-x + 3)^{\frac{1}{2}}[/tex]
This is a square root function. This graph only consists of x-values in the interval (3, ∞) because you cannot take the square root of a negative number. This function has a restricted domain.
C. [tex]g(x) = -(x + 8)^{3}[/tex]
This is a cubic function, so it has an interval of (-∞, ∞)
D. [tex]h(x) = (4x)^{2} - 5[/tex]
This is a quadratic function, which has no domain restriction.
Therefore, the correct answer is B.
