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

Answer 1

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 $f(x)=b^{x}$.

Answer 2

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.

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 $f(0)=1$.