Question

Determine if the function f is an exponential function. If so, identify the base. If not, why not?

Answer 1

Definition of Exponential Function : The exponential function with base b is denoted by f( x ) = b^x, where b > 0, b ≠ 1, and x is any real number.

An exponential function is a function of the form y = b^x.

where b is a positive number not equal to 1, and x is any real number.

Thus, exponential functions have a constant base;  the variable is in the exponent.

The number b is called the base of the exponential function.

The most important exponential function is when the base is the irrational number e. (Note:  e≈2.71828)

In this case, the function is also written as y = exp(x), and is called the natural exponential function.

The exponential function is f(x) = (1/e)^x → f(x) = (e^{- 1})^x → f(x) = (e)^{- x}.

Here base is e and its value is 2.718281828 . . . . > 0, and this number is called natural base, and x is any real number.

The function f(x) = (1/e)^x is called natural exponential function.

The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is exactly 1.

The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is also commonly defined as the base of the natural logarithm (using an integral to define the latter), as the limit of a certain sequence, or as the sum of a certain series.

Characteristics:      Such exponential graphs of the form  f (x) = bx  have certain characteristics in common:
Exponential functions are one-to-one functions.		•  graph crosses the y-axis at (0,1) •  when b > 1, the graph increases •  when 0 < b < 1, the graph decreases •  the domain is all real numbers •  the range is all positive real numbers (never zero) •  graph passes the vertical line test - it is a function •  graph passes the horizontal line test - its inverse is also a function. •  graph is asymptotic to the x-axis - gets very, very close to the x-axis but does not touch it or cross it.

Natural Exponential Function:

The function defined by  f (x) = e^x  is called the natural exponential function.

(e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler .)

Notice how the characteristics of this graph are similar to those seen above.

 This function is simply a "version" of
   where b >1.