# What is the inverse of the function f(x) = 7log_4(x+3) - 2? It is 7log_4 (x+3) - 2, if that clears any confusion.

Aug 5, 2016

$g \left(x\right) = {4}^{\frac{x + 2}{7}} - 3$

#### Explanation:

Calling $f \left(x\right) = 7 {\log}_{4} \left(x + 3\right) - 2$ we have

$f \left(x\right) = {\log}_{4} \left({\left(x + 3\right)}^{7} / {4}^{2}\right) = y$

Now we will proceed to obtain $x = g \left(y\right)$

${4}^{y} = {\left(x + 3\right)}^{7} / {4}^{2}$ or
${4}^{y + 2} = {\left(x + 3\right)}^{7}$
${4}^{\frac{y + 2}{7}} = x + 3$ and finally
$x = {4}^{\frac{y + 2}{7}} - 3 = g \left(y\right) = \left(g \circ f\right) \left(x\right)$

So $g \left(x\right) = {4}^{\frac{x + 2}{7}} - 3$ is the inverse of $f \left(x\right)$

Attached a plot with $f \left(x\right)$ in red and $g \left(x\right)$ in blue.