Let a and b be real number constants. An exponential function in x is a function that can be written in the form
f(x) = a ⋅ b^{x}
where a is nonzero, b is positive and b ≠ 1.
The constant a is the initial value of f (the value x = 0) and b is the base.
Let us consider the following functions,
For f(x) = x^{2}, the base is the variable x, and the exponent is the constant 2. So, f(x) is a monomial and it is power function.
For g(x) = 2^{x}, the base is the constant , the exponent is the variable x. g is an exponential function.
Note :
Exponential functions are defined and continuous for all real numbers.
Example :
Which of the following are exponential functions ?. For those that are exponential functions, state the initial value and the base. For those they are not, explain why not.
(a) y = x^{8}
(b) y = 3^{x}
(c) y = x^{√x}
(d) y = 7 ⋅ 2^{-x}
(e) y = 5 ⋅ 6^{π}
Solution :
(a) y = x^{8}
For the above function the base is the variable x, and the exponent is the constant 8. So, it is a monomial and it is power function.
(b) y = 3^{x}
For the above function the base is the constant , the exponent is the variable x. it is an exponential function.
Initial value(a) = 1 and base (b) = 3
(c) y = x^{√x}
Even though the above function has variable exponent, the base is not constant. So it is not a exponential function.
(d) y = 7 ⋅ 2^{-x}
This function exactly in the form a ⋅ b^{x}, so it is an exponential function.
Initial value (a) = 7 and base (b) = 2
(e) y = 5 ⋅ 6^{π}
The exponent is π, that is constant. So it is not an exponential function.
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