Question

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43. For the following functions, state whether they are growth or decay, then determine each function's growth/decay rate. (NOT MULTIPLE CHOICE)

a. f(x) = 0.3(1.6)^x
b. g(x) = 0.8^0.5x - 7
c. h(x) = 3(-5)^x
d. f(x) = e^(-0.45x) + 4

Answer 1

a) Exponential growth function: 60%

b) Exponential decay function: 20%

c) Neither

d) Exponential decay function: 45%

Explanation:

`\textsf{Exponential Growth}: \quad y=a(1+r)^x`

`\textsf{Exponential Decay}: \quad y=a(1-r)^x`

where:

• a = initial value (the amount before measuring growth or decay)
• r = growth/decay rate (in decimal form)

Part a

Given function:

`f(x) = 0.3(1.6)^x`

The function is an exponential growth function and the growth rate is:

`1+r=1.6 \implies r= 0.6 = 60\%`

Part b

Given function:

`g(x) = 0.8^(0.5x) - 7`

The function is an exponential decay function and the decay rate is:

`1-r=0.8 \implies r=0.2=20\%`

Part c

Given function:

`h(x) = 3(-5)^x`

This is neither an exponential growth or decay function as exponential functions cannot have negative bases.

Part d

Natural base exponential function

`y=ae^(rx)`

If a > 0 and r > 0, the function is an exponential growth function.

If a > 0 and r < 0, the function is an exponential decay function.

Given function:

`f(x) = e^(-0.45x) + 4`

As a = 1 > 0 and r = -0.45 < 0, the function is exponential decay function and the decay rate is:

`r=-0.45=45\%`