What is an exponential function?

Question

Can an exponential function be negative? Or they are always positive?
Give me examples please.

Is f(x) = -2^x an exponential function?

and isn't this a negative function?
Because if you graph it its graph is negative

Answer 1

Exponential function has the form y = a^x

Where x is the exponant.

The most used is y = e^x where e is a number e= 2.718...

and has the property that its derivative is also e^x

Exponential function is always positive

An other example is y =10^x which takes the values

for x=0 y = 1 , x=1 y=10 , x=2 y =100 and x = -1 y = 0.1

Answer 2

Definition Exponential Function

Answer 3

Well, this actually depends on how broad you want to make the definition of exponential function.

The most commonly encountered type of exponential function (and the one easiest to define without ambiguties) is f(x)=a^x, where a>0 and x is a real number. Indeed, many people restrict themselves specifically to considering _the_ exponential function, that is, f(x)=e^x (again, with x∈R). This type of exponential function is always positive.

However, we can define exponential functions more general than that. For instance, we can let f(x)=a^x with a<0, and x a rational number with odd denominator. This type of exponential function can be negative, and can be defined purely from the laws of exponents (specifically, a^(p/q) would be the real qth root of a, to the pth power. The requirement that q is odd ensures that this root exists). This is not frequently done, simply because of the difficulties with working with a domain which is that small -- however, you will frequently see power functions (read: functions of the form f(x)=x^a, instead of a^x) where the exponent is integer, or otherwise rational with odd denominator, and these power functions are usually understood to include negative numbers in the domain.

More frequently, we extend the exponential function by defining e^x by [n=0, ∞]∑x^n/n! for all _complex_ x, allowing us to sensibly interpret complex exponentiation. This function can be negative, in fact one of the celebrated identities is that e^(iπ)=-1.

Note that this definition also allows us to interpret exponentiation with any other nonzero base (including negative and even complex bases) to any complex power using the identity a^x = e^(x ln a), where ln a denotes the principal value of the complex logarithm. Note that this is often different, on the intersection of the domains, from the definition of exponentiation given by definition given in the third paragraph (for instance, (-1)^(1/3) would be -1 according to the third paragraph, but according to this method the value given is e^(iπ/3), which is (1+i√3)/2). Nonetheless, there is a connection, in that if these are instead considered as multivalued "functions", with the former definition taken to mean the pth power of _all_ the qth roots of a, and likewise with the current definition to mean e^(x ln a) for _all_ possible values of ln a, the set of solutions returned will be exactly the same (specifically, for (-1)^(1/3), we recieve -1, (1+i√3)/2, and (1-i√3)/2 as solutions).

So to recap:

positive base to real power: no
negative base to real power: yes, if exponentiation on negative bases is defined
any (nonzero) base whatsoever to complex power: yes

Answer 4

exponential function is one of the most important functions in mathematics. It is written as exp(x), where e equals approximately 2.71828183 and is the base of the natural logarithm.
The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.
The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.

As a function of the real variable x, the graph of y=ex is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x.

Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form ka^x, where a, called the base, is any positive real number. This article will focus initially on the exponential function with base e, Euler's number.

Answer 5

Exactly, you know I was only discussing that, this very night with a can of lager with a broken ring pull. I think the can of Export surmised that there are slightly more cons with exponential functions. AP: I'm not a fan of either but I'll go with bunnies, exponential of course.

Answer 6

All of the above is correct, unless, of course, you consider the fact that in the general expression y=a*n^x, 'a' could be negative.

Answer 7

exponential function has stuff like nine to the power of nine which is like saying nine times nine times nine times nine times nine times nine times nine times nine times nine equals whatever, so basically it's a shortened multiplication.