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What is the Natural Base Exponential Function?

What is the natural base exponential function?

Summary

  1. 'e' is the natural base, and is an irrational number approximately equal to 2.71828
  2. 'e' is NOT a variable -- it will always be equal to approximately 2.71828
  3. Anything to the 0 power is 1, so plugging in 0 for 'x' gives a y-value of 1
  4. Anything to the 1st power is just itself, so plugging in 1 for 'x' gives a y-value of 'e'
  5. The graph of f(x) = ex has a horizontal asymptote along the x-axis
  6. An asymptote is an imaginary line that a graph approaches but never reaches

Notes

    1. An exponential function is a function with a variable in the exponent
    2. In an exponential function 'a' cannot equal 0, and 'b' must be positive but cannot equal 1
    3. 'b' is the base of the exponential function
    1. For this exponential function, a = 2 and b = 5
    2. 5 is the base of the function
    1. A natural base exponential function is an exponential function whose base is 'e', the natural base
    1. 'e' is known as the 'natural base'
    2. That's why we call it a natural base exponential function!
    3. 'e' is an irrational number, so it is a decimal that goes on and on forever without repeating
    1. 'e' is the natural base, and is an irrational number approximately equal to 2.71828
    2. 'e' is NOT a variable -- the variable in this function is still 'x'
    3. 'e' is just the symbol we use for that specific irrational number
    4. 'e' is sort of like π -- it's a number we use so much that it gets a special symbol to represent it
    1. Plugging 0 in for 'x' into our function gives us f(0) = e0
    2. Remember, anything to the 0 power is 1, even 'e'
    3. So we get 1 as our value for f(x) when x = 0
    1. Plugging 1 in for 'x' into our function gives us f(1) = e1
    2. Remember, anything to the 1st power is just itself, even 'e'
    3. So we get 'e' as our value for f(x) when x = 1
    4. Remember, 'e' is approximately equal to 2.71828
    1. An asymptote is a line that a graph approaches but never crosses
    2. So as our x-values get more and more negative, the y-values will get closer and closer to 0 but will never actually reach 0
    3. That means there is nothing we can plug in for 'x' that will make our function equal 0
    1. The domain of a function is the set of all possible x-values, or inputs, for the function
    2. The range of a function is the set of all possible y-values, or outputs, for the function
    1. The domain of a function is the set of all possible x-values, or inputs, for the function
    2. From our graph we can see that the x-values continue on and on in both directions
    3. We can plug any real number in for 'x' and get a y-value out, so our domain is all real numbers
    1. The range of a function is the set of all possible y-values, or outputs, for the function
    2. Since we have a horizontal asymptote at y = 0, we know that our function will never equal 0
    3. And we can see from our graph that it will never go below 0
    4. There are no values we can plug in for 'x' that will give us an output that is not a positive real number
    5. So the range is the set of all positive real numbers