We are working with the function $f(x) = 3 \cdot 4^x$. Let's calculate each value step by step.

1. $f(-2)$

$f(-2) = 3 \cdot 4^{-2}$ Using the property $a^{-n} = \frac{1}{a^n}$: $f(-2) = 3 \cdot \frac{1}{4^2} = 3 \cdot \frac{1}{16} = \frac{3}{16}$ So, $f(-2) = \frac{3}{16} \approx 0.1875$

2. $f(12)$

$f(12) = 3 \cdot 4^{12}$ Calculate $4^{12}$: $4^{12} = (2^2)^{12} = 2^{24}$ Approximating $2^{24}$, we get: $2^{24} = 16777216, \quad 4^{12} = 16777216$ So, $f(12) = 3 \cdot 16777216 = 50331648$

3. $f(0)$

$f(0) = 3 \cdot 4^0$ Using the property $a^0 = 1$: $f(0) = 3 \cdot 1 = 3$

Final Results:

1. $f(-2) = \frac{3}{16}$ or approximately $0.1875$
2. $f(12) = 50331648$
3. $f(0) = 3$

Let me know if you'd like further clarification or need help with related problems!

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Related Questions:

1. What is the general behavior of exponential functions like $f(x) = 3 \cdot 4^x$?
2. How do changes in the base $b$ of $a \cdot b^x$ affect the function?
3. Can you rewrite $f(x) = 3 \cdot 4^x$ using logarithms?
4. What is the inverse of $f(x) = 3 \cdot 4^x$?
5. How do exponential growth functions relate to real-world phenomena?

Tip: Always check if the exponential base is positive and greater than 1 for meaningful growth or decay.