The form of the exponential function is
[tex]f(x)=a(b)^x[/tex]
a is the initial value (value f(x) at x = 0)
b is the growth/decay factor
Since the function has points (0, 6) and (3, 48), then
Substitute x by 0 and f(x) by 6 to find the value of a
[tex]\begin{gathered} x=0,f(x)=6 \\ 6=a(b)^0 \\ (b)^0=1 \\ 6=a(1) \\ 6=a \end{gathered}[/tex]
Substitute the value of a in the equation above
[tex]f(x)=6(b)^x[/tex]
Now, we will use the 2nd point
Substitute x by 3 and f(x) by 48
[tex]\begin{gathered} x=3,f(x)=48 \\ 48=6(b)^3 \end{gathered}[/tex]
Divide both sides by 6
[tex]\begin{gathered} \frac{48}{6}=\frac{6(b)^3}{6} \\ 8=b^3 \end{gathered}[/tex]
Since 8 = 2 x 2 x 2, then
[tex]8=2^3[/tex]
Change 8 to 2^3
[tex]2^3=b^3[/tex]
Since the powers are equal then the bases must be equal
[tex]2=b[/tex]
Substitute the value of b in the function
[tex]f(x)=6(2)^x[/tex]
The answer is:
The formula of the exponential function is
[tex]f(x)=6(2)^x[/tex]
