What is the inverse of the function $\newline$ $\begin{array}{l} g(x)=-\frac{2}{5} x+3 ? \\ g^{-1}(x)=\square \end{array}$

Full solution

Q. What is the inverse of the function

\newline

\begin{array}{l} g(x)=-\frac{2}{5} x+3 ? \\ g^{-1}(x)=\square \end{array}

Rewriting $g(x)$ as $y$ : To find the inverse of the function $g(x)$ , we need to switch the roles of $x$ and $y$ and then solve for $y$ . Let's start by rewriting $g(x)$ as $y$ : $\newline$ $y = -\left(\frac{2}{5}\right)x + 3$
Switching x and y: Now, we switch x and y to find the inverse: $x = -\left(\frac{2}{5}\right)y + 3$
Isolating y: Next, we want to isolate y on one side of the equation. To do this, we'll first move the constant term to the other side by subtracting $3$ from both sides: $\newline$ $x - 3 = -\left(\frac{2}{5}\right)y$
Getting rid of the coefficient: Now, we need to get rid of the coefficient $-\frac{2}{5}$ that is multiplying $y$ . We do this by multiplying both sides of the equation by the reciprocal of $-\frac{2}{5}$ , which is $-\frac{5}{2}$ : $\newline$ $\left(-\frac{5}{2}\right)(x - 3) = y$
Simplifying the equation: We can distribute $-\frac{5}{2}$ on the left side to simplify the equation: $\newline$ $y = \left(-\frac{5}{2}\right)x + \left(\frac{15}{2}\right)$
Finding the inverse function: We have now found the inverse function of $g(x)$ : $\newline$ $g^{-1}(x) = \left(-\frac{5}{2}\right)x + \left(\frac{15}{2}\right)$