DETERMINE WHETHER EACH GIVEN IS AN EXPONENTIAL FUNCTION

Determine which functions are exponential functions. For those that are not, explain why they are not exponential functions.

Problem 1 :

f(x) = 2^x + 7

Solution :

f(x) = 2^x + 7

Here, a = 1, b = 2

Since, the value of b is > 1. The given function f(x) is a exponential growth function.

Problem 2 :

g(x) = x²

Solution :

g(x) = x²

Here, a = 1, b = x

If it is a exponential function, the variable should be in the power. Since the power is 2, it must be a quadratic function.

Problem 3 :

h(x) = 1^x

Solution :

h(x) = 1^x

Here, a = 1, b = 1

Since, the function is a constant function.

Problem 4 :

f(x) = x^x

Solution :

f(x) = x^x

Here, a = 1, b = x

Since we have variable in the base and as well in power, it is not a exponential function. So, it is neither.

Problem 5 :

h(x) = 3 ⋅10^-x

Solution :

h(x) = 3 ⋅ 10^-x

h(x) = 3 ⋅ (1/10)^x

Here, a = 3, b = 1/10

Since, the value of b is 0 < b < 1. The given function h(x) is a exponential decay function.

Problem 6 :

f(x) = -3^{x + 1} + 5

Solution :

f(x) = -1(3^{x + 1} + 5)

a = 1, b = -3

Since, the value of b is 0 < b < 1. The given function f(x) is a exponential decay function.

Problem 7 :

g(x) = (-3)^{x + 1} + 5

Solution :

g(x) = (-3)^{x + 1} + 5

a = 1, b = -3

Since, the value of b is 0 < b < 1. The given function f(x) is a exponential decay function.

Problem 8 :

h(x) = 2x - 1

Solution :

h(x) = 2x - 1

Here, a = 1, b = 0

It is not in the form y = ab^x. So, it i s not an exponential function. 

Since it is in the form of y = ax + b, it is a linear function.