Answer
(a) $b^{x}=e^{xlnb}$
(b) $(-\infty,\infty)$
(c) $(0,\infty)$
(d) The general shape of the graph of the exponential function for each of the following cases is as depicted in below figure.
(i) $b>1 $
(ii) $b=1$
(iii) $0
Work Step by Step
(a) The equation that defines $b^{x}$ when b is a positive number and x is a real number must be refers to $ b^{x}=e^{xlnb}$.
(b) A function f is a rule that assigns to each element x in a set D exactly one element, called , in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function.
The domain for $f(x)=b^{x}(b>0)$ is for all real numbers. Hence, $(-\infty,\infty)$
(c) The number $f(x)=b^{x}$ is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of as x varies throughout the domain. Hence, $(0,\infty)$
(d) The general shape of the graph of the exponential function for each of the following cases is as depicted in below figure.
(i) $b>1$
(ii) $b = 1$
(iii) $0
