Given the exponential function: a(x) = p(1 + r)^x, what value for r will make the function a...

Question:

Given the exponential function: a(x) = p(1 + r)^x, what value for r will make the function a growth function?

Growth/Decay:

The exponential equation, {eq}A(x)=ab^x {/eq} is an exponenial equation where {eq}a {/eq} is the initial quantity. It

(i) represents the exponential growth if {eq}b>1 {/eq}.

(ii) represents the exponential decay if {eq}b<1 {/eq}.

The given function is:

$$A(x)= p(1+r)^x$$

Comparing this with {eq}A(x)=ab^x {/eq}, we get:

$$a=p ; \,\,\, b=1+r$$

We know that {eq}A(x)=ab^x {/eq} represents a growth function if {eq}b>1 {/eq}.

Substitute the value of {eq}b=1+r {/eq} in {eq}b>1 {/eq}.

Then we get:

$$1+r>1\\ \text{Subtracting 1 from both sides,} \\ r>0$$

Therefore, only the POSITIVE values of {eq}r {/eq} will make the function a growth function.