IDENTIFYING EXPONENTIAL FUNCTIONS

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)  =  x², 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  =  x⁸

(b)  y  =  3^x

(c)  y  =  x^√x

(d)  y  =  7 ⋅ 2^-x

(e)  y  =  5 ⋅ 6^π

Solution :

(a)  y  =  x⁸

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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