Do you mean [tex]f(x)=-3^{\sqrt x}[/tex], as in the negative of 3 to the power of √x ? Or [tex]f(x)=-\sqrt[3]{x}[/tex], as in the negative cube root of x ?
If you mean [tex]f(x)=-3^{\sqrt x}[/tex], then, going through each option:
• no: the square root component is defined only for x ≥ 0, for which we have √x ≥ 0 and hence f(x) < 0 for all x in the domain. √x is an increasing function, so the powers of 3 get successively larger, so f(x) would be always decreasing;
• no: as mentioned above, the domain would be all non-negative numbers;
• no: at minimum, x = 0 which gives f (0) = -3⁰ = -1, and because f(x) is decreasing over its entire domain, the range would be {y | y ≤ -1};
• no: again, a bit of ambiguity, but assuming you mean to say here [tex]y=3^{\sqrt x}[/tex], yes, f(x) would be a reflection of y across the x-axis;
• and finally, no: [tex]f(3)=-3^{\sqrt3}\approx-6.705[/tex]
so that only the fourth option would be true.
On the other hand, if you mean [tex]f(x)=-\sqrt[3]{x}[/tex], then:
• no: the function is always decreasing;
• yes: the domain is all real numbers;
• yes: the range is {y | -∞ < y < ∞};
• yes: f(x) is a reflection of y=∛x, also about the x-axis; and
• no: f (3) = -∛3 ≈ -1.442
I suspect you mean the second case, since it's a bit simpler to approach.
