We are supposed to explain why the given function can be expressed in form [tex]f(x)=b^{x}[/tex], when [tex]f(0)=1[/tex].
Since the given function is an exponential function we can express it as [tex]f(x)=a(b)^{x}[/tex].
Now let us substitute x=0 in the given function,
[tex]f(x)=b^{x}[/tex]
[tex]f(0)=b^{0}[/tex]
[tex]f(0)=1[/tex]
Let us substitute x=0 in our function.
[tex]f(0) = a(b)^{0}[/tex]
[tex]f(0) = a\cdot 1[/tex]
[tex]f(0)=a[/tex]
Upon substituting a=1 in our function we will get,
[tex]f(x) = 1(b)^{x}[/tex]
[tex]f(x) =(b)^{x}[/tex]
Therefore, our function f can be represented by the formula of the form [tex]f(x)=b^{x}[/tex].
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= [tex]f(x)= b^x[/tex], where [tex]b>1[/tex]
let b=1+n where n is any positive number.
Thus, [tex]f(x)=(1+n)^x[/tex] is an exponential function.( because its value is raising by the power of a constant [tex]1+n[/tex] )
And, for [tex]x=0[/tex], [tex]f(0)=(1+n)^0[/tex]⇒[tex]f(0)=1[/tex]
let [tex]n=1[/tex], [tex]f(x) =2^x[/tex] 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 [tex]f(0)=1[/tex].
