Question

Suppose f is a function with exponential growth and f(0)=1. Explain why f can be represented by a formula of the form


`f(x) = b^(x)`
for some b>1.

Answer 1

Answer: Because, here the rate of change is directly proportional to the value of f(x) and it is satisfying the condition f(0)=1.

Step-by-step explanation:

Since, Given function=
`f(x)= b^x`, where
`b>1`

let b=1+n where n is any positive number.

Thus,
`f(x)=(1+n)^x` is an exponential function.( because its value is raising by the power of a constant
`1+n` )

And, for
`x=0`,
`f(0)=(1+n)^0`⇒
`f(0)=1`

let
`n=1`,
`f(x) =2^x` after making its graph we found that the function is growing and growth rate is positive.

Similarly, function f will be growing exponential for the every value of n. where
`f(0)=1`.

Answer 2

We are supposed to explain why the given function can be expressed in form
`f(x)=b^(x)`, when
`f(0)=1`.

Since the given function is an exponential function we can express it as
`f(x)=a(b)^(x)`.

Now let us substitute x=0 in the given function,

`f(x)=b^(x)`

`f(0)=b^(0)`

`f(0)=1`

Let us substitute x=0 in our function.

`f(0) = a(b)^(0)`

`f(0) = a\cdot 1`

`f(0)=a`

Upon substituting a=1 in our function we will get,

`f(x) = 1(b)^(x)`

`f(x) =(b)^(x)`

Therefore, our function f can be represented by the formula of the form
`f(x)=b^(x)`.