Question

Which is an exponential decay function?

f start bracket x end bracket equals Three-fourths start bracket Seven-fourths end bracket superscript x
f start bracket x end bracket equals two-third start bracket Four-fifths end bracket superscript negative x
f start bracket x end bracket equals Three-halves start bracket Eight-sevenths end bracket superscript negative x
f start bracket x end bracket equals one-third start bracket negative nine-halves end bracket superscript x

Answer 1

The third one:

f(x) = (3/2)(8/7)^{-x} = (3/2)(7/8)^x, and 7/8 < 1 so the function decreases (exponential decay).

(For comparison: 1) 7/4 > 1 → growth; 2) (4/5)^{-x} = (5/4)^x > 1 → growth; 4) (-9/2)^x is not a positive-base exponential and is not a decay function.)