Question

In exponential growth functions, the base of the exponent must be greater than 1. How would the function change if the base of the exponent were 1? How would the function change if the base of the exponent were between 0and 1?

Answer 1

Explanation:

Formula for exponential growth function is written as:

$A_{t}=A_{0}(1+x)^t$, where $A_{t},A_{0}$ are final value and initial value of any substance respectively.

x is the rate at which the substance is increasing.

t = total time for which growth is calculated.

So, if we replace ,$(1+x)^t by V^t$ then formula for exponential growth functions will be written as

$A_{t}=A_{0}V^t$,

(a) If , Base =V=1,

$A_{t}=A_{0}1^t=A_{0}$

The value of the function will remain constant value that is a real number.

(b) If 0<V<1, then

the function becomes decreasing in the interval (0, ∞]. Function almost touches the x axis ,as Asymptote becomes y=0.

Answer 2

We will see that:

• For a base equal to 1: We have a constant function.
• For a base between 0 and 1: We have an exponential decay.

What does the base in an exponential function?

A general exponential function can be written as:

f(x) = A*(r)^x

Where r is the base.

What happens if the base is 1?

We will have:

f(x) = A*(1)^x = A

So we have a constant function.

What happens if the base is between 0 and 1?

If 0 < r < 1

We will have that:

$r^x > r^{x +1}$

Then for:

f(x) = A*(r)^x

We will have that:

f(x) > f(x + 1)

Meaning that as x increases, the function decreases, then we have an exponential decay.

If you want to learn more about exponential functions, you can read:

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