Let's break down the given function to evaluate $f(-2.2)$.

The function is: $f(x) = 2.4(4)^{-x+1} + 1.2$ You are asked to find $f(-2.2)$.

Step 1: Substitute $x = -2.2$ into the function

$f(-2.2) = 2.4(4)^{-(-2.2) + 1} + 1.2$

Simplify the exponent: $-(-2.2) + 1 = 2.2 + 1 = 3.2$

So now we have: $f(-2.2) = 2.4(4)^{3.2} + 1.2$

Step 2: Calculate $(4)^{3.2}$

Using a calculator: $4^{3.2} \approx 85.7939$

Step 3: Multiply by 2.4

$2.4 \times 85.7939 \approx 205.9054$

Step 4: Add 1.2

$f(-2.2) = 205.9054 + 1.2 = 207.1054$

Final Answer:

$f(-2.2) \approx 207.1054$

Would you like more details on any step?

Here are 5 questions to deepen your understanding:

1. What are the properties of exponents that were used here?
2. How do we handle negative exponents and negative values for $x$?
3. What would change if the base 4 in the exponent were replaced with a different number?
4. How does rounding affect the final result in functions like these?
5. What role does the constant 1.2 play in the overall behavior of the function?

Tip: When working with exponents, pay close attention to negative signs, as they can change the behavior of the function significantly.