Answer
[tex]f(x)=3(2)^x[/tex]
Explanation
To solve this, we re using the standard exponential equation: [tex]f(x)=ab^x[/tex]
Also, remember that [tex]f(x)=y[/tex]
We know that our exponential function goes through the points (1, 6) and (2, 12), so we are going to replace the points in our stander exponential equation and solve for [tex]a[/tex] and [tex]b[/tex]:
For (1, 6)
[tex]x=1[/tex] and [tex]y/f(x)=6[/tex], so:
[tex]f(x)=ab^x[/tex]
[tex]6=ab^1[/tex]
[tex]6=ab[/tex]
[tex]a=\frac{6}{b}[/tex] equation (1)
For (2, 12)
[tex]x=2[/tex] and [tex]y/f(x)=12[/tex], so:
[tex]f(x)=ab^x[/tex]
[tex]12=ab^2[/tex] equation (2)
Replace equation (1) in equation (2) and solve for b
[tex]12=ab^2[/tex]
[tex]12=(\frac{6}{b}) b^2[/tex]
[tex]12=6b[/tex]
[tex]b=\frac{12}{6}[/tex]
[tex]b=2[/tex] equation (3)
Replace equation (3) in equation (1) to find a
[tex]a=\frac{6}{b}[/tex]
[tex]a=\frac{6}{2}[/tex]
[tex]a=3[/tex]
Now we can complete our exponential function:
[tex]f(x)=ab^x[/tex]
[tex]f(x)=3(2)^x[/tex]
We can conclude that the exponential function that goes through the points (1, 6) and (2, 12) is [tex]f(x)=3(2)^x[/tex]
