Growth vs Decay Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Classify each function as growth or decay, and find its value at

x=3

: (a)

f(x)=4\cdot2^x

, (b)

g(x)=100\cdot(0.5)^x

1
(a) Base $b=2>1$ : exponential growth. $f(3)=4\cdot8=32$ .
2
(b) Base $b=0.5$ , $0<0.5<1$ : exponential decay. $g(3)=100\cdot(0.5)^3=100\cdot0.125=12.5$ .
3
Interpretation: (a) doubles with each unit increase; (b) halves with each unit increase.

Answer

(a) Growth,

f(3)=32

; (b) Decay,

g(3)=12.5

For

y=a\cdot b^x

with

a>0

: if

b>1

the function grows exponentially; if

0<b<1

it decays exponentially. The base

b

determines direction; the coefficient

a

sets the initial value.

About Growth vs Decay

Exponential growth occurs when a quantity multiplies by a factor $> 1$ repeatedly; exponential decay when it multiplies by a factor between 0 and 1.

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