what are the properties of exponential function?
Since this is not my area of expertise, I searched Google under the key words "exponential function properties" to get these possible sources:
http://en.wikipedia.org/wiki/Exponential_function
(Broken Link Removed)
http://home.scarlet.be/~ping1339/exp.htm
http://tutorial.math.lamar.edu/Classes/Alg/ExpFunctions.aspx
http://www.rapidlearningcenter.com/mathematics/pre-calculus/09-Exponential-Functions-Graphs-and-Models.html
In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search. Also see http://hanlib.sou.edu/searchtools/.
I hope this helps. Thanks for asking.
Well, let me find my trusty clown calculator to give you the answer! Ah, here it is! So, the properties of an exponential function are quite simple, yet powerful. First, exponential functions have a base, let's call it 'b', which is a positive number. Second, they have the form f(x) = b^x, where 'x' is the input variable. Third, they exhibit rapid growth or decay depending on whether the base 'b' is greater than or less than 1. And finally, they have this sneaky little property called the exponential growth or decay factor, which determines how fast the function increases or decreases. So, in a nutshell, exponential functions are like little math rockets that can either skyrocket or nosedive, depending on their base and the value of 'x'. Isn't math funny? Well, that's debatable...
The properties of exponential functions can be summarized as follows:
1. Domain and Range: The domain of an exponential function is all real numbers, and the range depends on the base of the function. If the base is greater than 1, the range is (0, infinity). If the base is between 0 and 1, the range is (0, infinity).
2. Growth and Decay: Exponential functions can either grow or decay. If the base is greater than 1, the function will grow and become increasingly large as the input increases. If the base is between 0 and 1, the function will decay and become increasingly small as the input increases.
3. Asymptote: The graph of an exponential function has a horizontal asymptote at y = 0. This means that as the x-values approach negative or positive infinity, the y-values get closer and closer to 0.
4. Increasing and Decreasing: Exponential functions are always increasing or always decreasing, depending on the base. If the base is greater than 1, the function is increasing. If the base is between 0 and 1, the function is decreasing.
5. The Rule of Exponents: Exponential functions follow the rule of exponents, which states that for any positive real number a and any real number x and y, a^x * a^y = a^(x+y). This property allows for simplification and manipulation of exponential expressions.
The properties of an exponential function are as follows:
1. Increasing or Decreasing: An exponential function can either be increasing or decreasing. If the base of the exponential function (denoted by "a") is greater than 1, the function will be increasing. Conversely, if the base is between 0 and 1 (exclusive), the function will be decreasing.
2. Asymptote: Exponential functions have a horizontal asymptote. If the base of the function is greater than 1, the asymptote will be at y = 0. If the base is between 0 and 1 (exclusive), the asymptote will be a positive value.
3. Domain and Range: The domain of exponential functions is all real numbers, since any real number can be raised to any power. The range, however, depends on whether the function is increasing or decreasing. If it is increasing, then the range will be all positive real numbers. If it is decreasing, the range will be all positive numbers approaching zero.
4. Intercepts: Exponential functions rarely have x-intercepts, except when the base is less than 1 and the exponent is a positive even number. As for y-intercepts, they occur when x = 0. The y-intercept is always equal to the initial value or the value of the function when x = 0.
To find the properties of a specific exponential function, you need to be given the function's equation, usually in the form y = a * b^x, where "a" represents the initial value, and "b" represents the base. Then you can apply these properties to analyze how the function behaves.