Answer:
C. The growth factor of g is twice the growth factor of f.
Step-by-step explanation:
Let's find the growth factor of g(x) by getting its equation. To do it, we are using the standard exponential equation:
[tex]y=a(b+1)^x[/tex]
where
[tex]a[/tex] is the initial value
[tex]b[/tex] is the growth factor
We know form our graph that g(x) passes throughout (0, 3), so [tex]x=0[/tex] and [tex]y=3[/tex].
Replacing values
[tex]3=a(b+1)^0[/tex]
[tex]3=a(1)[/tex]
[tex]a=3[/tex]
We also know from our graph the g(x) passes throughout (1, 12), so [tex]x=1[/tex] and [tex]y=12[/tex].
Replacing values
[tex]y=a(b+1)^x[/tex]
[tex]12=3(b+1)^1[/tex]
[tex]12=3(b+1)[/tex]
[tex]b+1=\frac{12}{3}[/tex]
[tex]b+1=4[/tex]
[tex]b=4-1[/tex]
[tex]b=3[/tex]
The growth factor of g(x) is 4.
Now, to find the growth factor of f(x), we just need to equate 1+b with [tex]\frac{5}{2}[/tex] and solve for b:
[tex]1+b=\frac{5}{2}[/tex]
[tex]b=\frac{5}{2} -1[/tex]
[tex]b=\frac{3}{2}[/tex]
[tex]b=1.5[/tex]
Finally, we can divide the growth factor of g(x) by the growth factor of f(x) to find how many times bigger is the growth factor of g(x):
[tex]\frac{3}{1.5} =2[/tex]
We can conclude that the growth factor of g is twice the growth factor of f.
