Sebastian writes the recursive formula f(x+1) = 4f(x) to
represent a geometric sequence whose second term is 12.
which explicit formula can be used to model the same?
sequence?
f(x) = 12(4)
f(x) = 3(4)x - 1
• f(x) = 4(12)*
f(x) = 4(3)* - 1

Step-by-step explanation leading to

Given: f(x+1) = 4*f(x)  [recursive] (1)  

and second term = 12

Need explicit formula  [non-recursive]

Solution:

A recursive formula is one which has the same term on both sides, but with different arguments, as in (1).

To visualize what (1) does, we set (arbitrarily) f(1)=1,  

so that using (1), we have

f(2) = 4*f(1) = 4

f(3) = 4*f(2) = 16

f(4) = 4*f(3) = 64,  

essentially we have a geometric series with common ratio of 4.

An explicit formula is f(x) = f(1)*(4^(x-1))   (2)

where the first term is interpreted as follows : f(1)=1.

We are given that the second term is 12, so we get

f(2) = f(1)*(4^(2-1), using equation (2)

substitute f(2) = 12 and Simplify:

12 = f(1)*(4^1) = 4*f(1) (3)

solve (3) for f(1)

f(1) = 12/4 = 3  (4)

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: