Question

Which properties are present in a table that represents an exponential function in the form mc029-1.jpg when mc029-2.jpg?

Answer 1

Answer:

where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form {\displaystyle f(x)=ab^{cx+d}} is also an exponential function, as it can be rewritten as

{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}

As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:

{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}

For b = 1 the real exponential function is a constant and the derivative is zero because {\displaystyle \log _{e}b=0,} for positive a and b > 1 the real exponential functions are monotonically increasing (as depicted for b = e and b = 2), because the derivative is greater than zero for all arguments, and for b < 1 they are monotonically decreasing (as depicted for b = 1/2), because the derivative is less than zero for all arguments.