Question

the domain of an exponential function is all real numbers. the range of an exponential function is positive real numbers. the graph of an exponential function has a y - intercept only. the graph is throughout its domain and has at the x - axis because the value ( b^{x} ) can get very close to 0 but never reach it.

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.