what is the range of the function ( y = e^{4x} )?\n( ylt0 )\n( ygt0 )\n( ylt4 )\n( ygt4 )

Explanation:

Step1: Analyze the exponential function properties

The general form of an exponential function is (y = a^{x}), where (a>0,a\neq1). For the function (y = e^{u}), the base (e\approx2.718>0). The value of (e^{u}) is always positive for any real - valued (u). Here (u = 4x), and (x\in R).

Step2: Determine the range

Since (4x) can take any real value (because (x\in R)), and (y = e^{4x}), and (e^{t}>0) for all (t\in R) (let (t = 4x)). As (x\to-\infty), (4x\to-\infty) and (\lim_{x\to-\infty}e^{4x}=0) (approaching (0) but never reaching (0)), and as (x\to+\infty), (4x\to+\infty) and (e^{4x}\to+\infty).

Answer:

(y>0)