Question

introduction to exponential functions
according to the mini - lesson, which of the following are true regarding ( f(x)=a(b)^{x} )? check all that apply. assume ( a>0 ).
the domain of the exponential functions is ( x>0 ).
the range of the exponential functions is ( f(x)>0 ).
the horizontal asymptote is the line ( y = 0 ).
the horizontal asymptote is the point ( (0,a) ).
the horizontal asymptote is the line ( x = 0 ).
the domain of the exponential functions is all real numbers.
the range of the exponential functions is all real numbers.

Explanation:

• Domain: For an exponential function \(y = a(b)^{x}\) (\(a>0\)), \(x\) can be any real number. So the domain is all real numbers, not \(x > 0\).
• Range: Since \(a>0\) and \(b^{x}>0\) for \(b>0,b

eq1\), then \(y=a(b)^{x}>0\). The range is \(y>0\), not all real numbers.

• Horizontal Asymptote: As \(x

ightarrow-\infty\) (if \(b > 1\)) or \(x
ightarrow\infty\) (if \(0 < b<1\)), \(y=a(b)^{x}
ightarrow0\). The horizontal asymptote is \(y = 0\). A horizontal asymptote is a line (not a point), and \(x = 0\) is a vertical line (not a horizontal asymptote).

Answer:

• The Range of the exponential functions is \(f(x)>0\).
• The Horizontal Asymptote is the line \(y = 0\).
• The Domain of the exponential functions is All Real Numbers.