use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation…

Explanation:

Step1: Analyze the transformation of the function

The parent function is (y = e^{x}). For the function (h(x)=e^{2x}+6), compared with (y = e^{x}), there is a horizontal compression by a factor of (\frac{1}{2}) (due to the (2x) inside the exponent) and a vertical shift up by (6) units.

The general form of an exponential function is (y = a\cdot e^{bx}+c). For the function (y = e^{x}), the asymptote is (y = 0).

Step2: Find the asymptote

For the function (h(x)=e^{2x}+6), using the transformation rules. When we shift the function (y = e^{x}) up by (6) units, the equation of the asymptote changes. If (y = f(x)) has an asymptote (y = k), then (y=f(x)+m) has an asymptote (y=k + m). Since the asymptote of (y = e^{x}) is (y = 0), for (h(x)=e^{2x}+6), the asymptote is (y = 6)

Step3: Determine the domain

The domain of an exponential function (y = a\cdot e^{bx}+c) (where (a\neq0), (b\neq0)) is all real numbers. For (h(x)=e^{2x}+6), since we can substitute any real - value of (x) into the function (h(x)), the domain is ((-\infty,\infty))

Step4: Determine the range

We know that (e^{2x}>0) for all (x\in R). Then (h(x)=e^{2x}+6>0 + 6). So the range is ((6,\infty))

Answer:

Asymptote: (y = 6); Domain: ((-\infty,\infty)); Range: ((6,\infty))