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

Full solution

Q. What is the inverse of the function

\newline

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

Replace with $y$ : To find the inverse of the function $g(x) = -(\frac{2}{3})x - 5$ , we first replace $g(x)$ with $y$ to make the equation easier to work with. $\newline$ $y = -(\frac{2}{3})x - 5$
Swap x and y: Next, we swap x and y to find the inverse function. This means we replace $y$ with $x$ and $x$ with $y$ in the equation. $x = -\left(\frac{2}{3}\right)y - 5$
Isolate y term: Now, we solve for $y$ to get the inverse function. First, we add $5$ to both sides of the equation to isolate the term with $y$ on one side. $x + 5 = -(\frac{2}{3})y$
Multiply by $-\frac{3}{2}$ : Next, we multiply both sides of the equation by $-\frac{3}{2}$ to solve for $y$ . $y = -\frac{3}{2} \times (x + 5)$
Simplify equation: We distribute the $-\frac{3}{2}$ across the parentheses to simplify the equation. $\newline$ $y = -\frac{3}{2} \times x - \frac{3}{2} \times 5$
Simplify constant term: Finally, we simplify the constant term $-\frac{3}{2} \times 5$ . $\newline$ $y = -\frac{3}{2} \times x - \frac{15}{2}$