Find the inverse of the function

Question

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.

Answer 1

Hint

Try to express $x$ with $y$ and you find

$$y=f(x)=-2\cdot4^{2(x-3)}-1\iff x=\frac12\log_4\left(-\frac{y+1}2\right)+3$$

Answer 2

The inverse of a function is found by substituting y for x and vice versa. So, we alter our equation to look like this:

$x = -2 * 4^{2(y-3)} - 1$

and solve for y.

$x + 1 = -2 * 4^{2(y-3)}$

$\frac{x + 1}{-2} = 4^{2(y-3)}$

$log_{4}(\frac{x + 1}{-2}) = 2(y-3)$

$\frac{log_{4}(\frac{x + 1}{-2})}{2} = y - 3$

$y = \frac{log_{4}(\frac{x + 1}{-2})}{2} + 3$

Therefore, $f(x)^{-1} = \frac{log_{4}(\frac{x + 1}{-2})}{2} + 3$