# Question #2a1a4

Mar 25, 2017

The inverse function is: ${g}^{-} 1 \left(x\right) = \frac{21}{3 - x} - 7$. See explanation.

#### Explanation:

To find the inverse function you have to transform the formula of $g \left(x\right)$ to calculate the value of $x$ in terms of $y$:

$y = \frac{3 x}{x + 7}$

$y = \frac{3 x + 21}{x + 7} - \frac{21}{x + 7}$

$y = 3 - \frac{21}{x + 7}$

$\frac{21}{x + 7} = 3 - y$

$\frac{1}{x + 7} = \frac{3 - y}{21}$

Now we can change the fractions to their reciprocals so that $x$ appears in the numerator:

$x + 7 = \frac{21}{3 - y}$

$x = \frac{21}{3 - y} - 7$

Now we can "rename" the argument and the value to write the formula using the standard convention ($x$ as the argument and $y$ as the function's value)

$y = \frac{21}{3 - x} - 7$

Finally we can write the answer: