Questions: f(x)=1/2 * 4^(x+7)-6 f^(-1)(x)=

Solution Steps

To find the inverse of the function \( f(x) = \frac{1}{2} \cdot 4^{x+7} - 6 \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \) to solve for \( y \) in terms of \( x \).
3. Isolate \( y \) to find the inverse function \( f^{-1}(x) \).

We start with the function defined as: \[ f(x) = \frac{1}{2} \cdot 4^{x+7} - 6 \]

To find the inverse, we set \( f(x) \) equal to \( y \): \[ y = \frac{1}{2} \cdot 4^{x+7} - 6 \] Swapping \( x \) and \( y \) gives us: \[ x = \frac{1}{2} \cdot 4^{y+7} - 6 \]

Rearranging the equation to isolate \( y \): \[ x + 6 = \frac{1}{2} \cdot 4^{y+7} \] Multiplying both sides by 2: \[ 2(x + 6) = 4^{y+7} \] Taking the logarithm base 4 of both sides: \[ y + 7 = \log_4(2(x + 6)) \] Thus, we can express \( y \) as: \[ y = \log_4(2(x + 6)) - 7 \]

Using the change of base formula, we can rewrite this as: \[ y = \frac{\log(2(x + 6))}{\log(4)} - 7 \] Since \( \log(4) = 2\log(2) \), we can simplify further: \[ y = \frac{1}{2} \log(2(x + 6)) - 7 \]

Final Answer