Question

Which set of ordered pairs could be generated by an exponential function?
(0, 0), (1, 1), (2, 8), (3, 27)
(0, 1), (1, 2), (2, 5), (3, 10)
(0, 0), (1, 3), (2, 6), (3, 9)
(0, 1), (1, 3), (2, 9), (3, 27)

Answer 1

An exponential function (of the form y = a·b^x) has y(0) = a ≠ 0 and constant successive ratio b = y(x+1)/y(x).

Check the sets:
- (0,0),...: y(0)=0 so a=0 ⇒ function would be zero everywhere, so no.
- (0,1),(1,2),(2,5),(3,10): ratios 2/1 = 2, 5/2 = 2.5 → not constant.
- (0,0),(1,3),(2,6),(3,9): y(0)=0 so same problem as first.
- (0,1),(1,3),(2,9),(3,27): ratios 3/1 = 3, 9/3 = 3, 27/9 = 3 → constant (a=1, b=3).

Thus the fourth set is exponential.