select the correct answer. consider the graph of the function f(x)=e^x. which statement describes a key…

Explanation:

Step1: Recall properties of exponential - functions

The parent function is (y = e^{x}), which has a range of (y>0), a domain of all real numbers ((x\in(-\infty,\infty))), a (y) - intercept at ((0,1)), and a horizontal asymptote at (y = 0). The function (g(x)=e^{x}-7) is a vertical translation of the function (y = e^{x}) down 7 units.

Step2: Analyze the range

For (y = e^{x}), (y>0). For (g(x)=e^{x}-7), we get (y> - 7) by subtracting 7 from each part of the inequality (y = e^{x}>0). So option A is incorrect.

Step3: Analyze the domain

The domain of the exponential function (y = e^{x}) is all real numbers, and a vertical translation does not change the domain. So the domain of (g(x)=e^{x}-7) is (x\in(-\infty,\infty)), and option B is incorrect.

Step4: Analyze the (y) - intercept

To find the (y) - intercept, we set (x = 0). Then (g(0)=e^{0}-7=1 - 7=-6). So the (y) - intercept is ((0,-6)), and option C is incorrect.

Step5: Analyze the horizontal asymptote

The horizontal asymptote of (y = e^{x}) is (y = 0). When we translate (y = e^{x}) down 7 units to get (g(x)=e^{x}-7), the horizontal asymptote also moves down 7 units. So the horizontal asymptote of (g(x)) is (y=-7).

Answer:

D. horizontal asymptote of (y=-7)