For the function 𝑔g, 𝑔(0)=94g(0)=94, and for each increase in 𝑥x by 11, the value of 𝑔(𝑥)g(x) decreases by 75%75%. What is the value of 𝑔(3)g(3)?

1. Break Down the Problem

We need to find the value of $g(3)$ given the initial value $g(0) = 94$ and the information that for each increase in $x$ by 11, $g(x)$ decreases to 25% of its previous value.

2. Relevant Concepts

The function experiences exponential decay. If $g(x)$ is the value of the function at a specific $x$, then: $g(x + 11) = 0.25 \cdot g(x)$

3. Analysis and Detail

1. At $g(0) = 94$, we set $x = 0$.
2. Thus, $g(11) = 0.25 \cdot g(0) = 0.25 \cdot 94 = 23.5$.
3. Now calculate $g(22)$: $g(22) = 0.25 \cdot g(11) = 0.25 \cdot 23.5 = 5.875$
4. Next, calculate $g(33)$: $g(33) = 0.25 \cdot g(22) = 0.25 \cdot 5.875 = 1.46875$

Since $g(x)$ changes significantly for each 11-unit increase, we need to interpolate to find $g(3)$. We can estimate it using decay from $g(0)$ to $g(11)$:

• The distance from 0 to 11 is 11 units, while we need the value at 3, so: $\text{Fraction of the way to } g(11) = \frac{3}{11}$

4. Verify and Summarize

1. We'll find the value of the function at some intermediate distance: $g(3) \approx g(0) + \left( \frac{3}{11} \times (g(11) - g(0)) \right)$ $g(11) - g(0) = 23.5 - 94 = -70.5$ Thus, $g(3) \approx 94 + \left( \frac{3}{11} \times (-70.5) \right) = 94 - 19.25 \approx 74.75$

Final Answer

The value of $g(3)$ is approximately 74.75.