the domain of an exponential function is all real numbers. the range of an exponential function is positive…

Explanation:

Step1: Analyze the behavior of the exponential function (y = b^{x}(b>1))

For any real - number (x), the function (y = b^{x}) is well - defined. As (x) varies over all real numbers ((x\in(-\infty,\infty))), the function is continuous. When (x) increases, (y=b^{x}) increases. When (x) approaches (-\infty), (y = b^{x}=\frac{1}{b^{-x}}) approaches (0) (but (y>0)), and when (x = 0), (y=b^{0}=1).

Step2: Determine the nature of the graph

Since the derivative of (y = b^{x}) (using the formula (\frac{d}{dx}(b^{x})=b^{x}\ln b) and (b > 1) implies (\ln b>0)), and (b^{x}>0) for all (x\in R), the function (y = b^{x}) has a positive derivative for all (x\in R). So the function is increasing for all (x) in its domain. Also, as (x\to-\infty), (y = b^{x}\to0), so the (x) - axis ((y = 0)) is a horizontal asymptote.

Answer:

The graph is (\text{increasing}) throughout its domain and has (\text{a horizontal asymptote}) at the (x) - axis.