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QuestionThe town's population decreases by 8%8\% yearly. If P(x)=0.92xP(x)=0.92x, find (P∘P)(x)(P \circ P)(x) in exact form.

Studdy Solution

STEP 1

Assumptions1. The population decreases at a rate of8% per year. . The population at a certain fixed time is represented by xx.
3. The population1 year later is represented by (x)=0.92x(x)=0.92x.
4. We need to find (∘)(x)( \circ)(x), which represents the population years later.

STEP 2

The function (∘)(x)( \circ)(x) represents the composition of the function (x)(x) with itself. This means we apply the function (x)(x) twice.

STEP 3

First, we apply the function (x)(x) to xx. This gives us the population1 year later.
(x)=0.92x(x) =0.92x

STEP 4

Now, we apply the function (x)(x) to the result of (x)(x), which is 0.92x0.92x. This gives us the population2 years later.
(∘)(x)=(P(x))=(0.92x)( \circ)(x) =(P(x)) =(0.92x)

STEP 5

Substitute 0.92x0.92x into the function (x)(x) to find the population2 years later.
(∘)(x)=0.92⋅(0.92x)( \circ)(x) =0.92 \cdot (0.92x)

STEP 6

Perform the multiplication to find the exact form of (∘)(x)( \circ)(x).
(∘)(x)=0.92⋅0.92x=0.8464x( \circ)(x) =0.92 \cdot0.92x =0.8464xSo, (∘)(x)=0.8464x( \circ)(x) =0.8464x.

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