Question

How do you identify equations as exponential growth, exponential decay, linear growth or linear decay `f(x) = 3 (1/2) ^2`?

Answer 1

The function that you provided is just a constant: 0.75.

However, we can think about what each of those terms means:

Exponential Growth / Decay:
`f(x+1) = a * f(x)`
for some number `a` that we call the base. If `f(x+1)>f(x)`, it's growth. Otherwise, it's decay. This means that if `a > 1`, we have a growth and if `a < 1`, we have a decay. Functions of this type look like the following:
`f(x) = c * a^x`
If the number you provided is part of an exponential decay, it may look like this
`f(x) = 3 * (1/2)^x`

If the number you provided is part of an exponential growth, it may look like this
`f(x) = (1/2)^2 3^x`

Linear Growth / Decay:
`f(x+1) = a + f(x)`
for some number `a` that we call the slope. If `f(x+1) > f(x)`, it's growth. Otherwise, it's decay. This means that if `a >0`, we have a growth and if `a< 0` we have a decay. Functions of this type look like the following:
`f(x) = a x + b`

If the number you provided was part of a linear growth, it may look like this
`f(x) = (1/2)^2 x`
If the number you provided was part of a linear decay, it may look like this
`f(x) = - x * (1/2)^2 + 1`