Question

Which function represents exponential decay?

f(x) = One-half (three-halves) Superscript x
f(x) = One-half (negative three-halves) Superscript x
f(x) = 4(negative two-thirds) Superscript x
f(x) = 4(two-thirds) Superscript x

Answer 1

Exponential decay occurs when the base b satisfies 0 < b < 1.

Evaluate the bases:
- (1/2)(3/2)^x → base = 3/2 (>1) → growth
- (1/2)(-3/2)^x → |base| = 3/2 (>1) → grows in magnitude (alternates sign)
- 4(-2/3)^x → |base| = 2/3 (<1) → magnitude decays but values alternate sign (not a standard positive exponential)
- 4(2/3)^x → base = 2/3 (0 < b < 1) → exponential decay

So the clear exponential decay is f(x) = 4(2/3)^x. (4(−2/3)^x decays in absolute value but alternates sign.)