Question

if the graph of an exponential function does not cross the x - axis, what does this imply about the asymptote?
it has a vertical asymptote.
it has no asymptote.
it has a horizontal asymptote at ( y = 0 ).
it has a horizontal asymptote at ( y = 1 ).

Explanation:

Step1: Recall the properties of exponential functions

The general form of an exponential function is \(y = a\cdot b^{x}+c\), where \(a
eq0\), \(b> 0\), \(b
eq1\). The horizontal asymptote of the exponential function \(y = a\cdot b^{x}+c\) is \(y = c\). When the graph of an exponential function does not cross the \(x -\)axis (\(y = 0\)), it means that the function approaches a non - zero horizontal line as \(x\to\pm\infty\).

Step2: Analyze the horizontal asymptote

For an exponential function \(y = a\cdot b^{x}+c\), if \(y>0\) for all \(x\) (the graph does not cross the \(x -\)axis), when \(a>0\) and \(b > 1\) or \(0 < b<1\), as \(x\to-\infty\) (if \(b>1\)) or \(x\to+\infty\) (if \(0 < b < 1\)), \(y=a\cdot b^{x}+c\to c\). Since \(y>0\) for all \(x\), the horizontal asymptote \(y = c\) where \(c\geq0\). In the case of the basic exponential function \(y = a\cdot b^{x}\) (\(c = 0\) is not possible as the graph would cross the \(x -\)axis in the limit as \(x\to-\infty\) for \(b>1\) or \(x\to+\infty\) for \(0 < b<1\) if \(a>0\)). For a non - crossing \(x -\)axis exponential function \(y=a\cdot b^{x}+c\) (\(a
eq0\), \(b>0,b
eq 1\)), the horizontal asymptote is \(y = 0\) is not correct (because if \(y = 0\) were a horizontal asymptote, the function would get arbitrarily close to \(y = 0\) and cross it for some large \(|x|\) in some cases). The function \(y=a\cdot b^{x}+c\) has no vertical asymptotes (since the domain of \(y = a\cdot b^{x}+c\) is all real numbers, \(x\in(-\infty,\infty)\)). And we know that an exponential function \(y=a\cdot b^{x}+c\) has a horizontal asymptote.

Answer:

It has a horizontal asymptote at \(y = 0\)