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$What is the inverse of the function {:[g(x)=-(2)/(3)x-5?],[g^(-1)(x)=]:}$

What is the inverse of the function $g(x) = -\frac{2}{3}x - 5$ ? $g^{-1}(x) =$

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Identify inverse functions

Full solution

Q. What is the inverse of the function

g(x) = -\frac{2}{3}x - 5

?

g^{-1}(x) =

Replace $g(x)$ with $y$ : To find the inverse of the function $g(x) = -\left(\frac{2}{3}\right)x - 5$ , we first replace $g(x)$ with $y$ for easier manipulation. $\newline$ $y = -\left(\frac{2}{3}\right)x - 5$
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. $\newline$ $x = -\left(\frac{2}{3}\right)y - 5$
Solve for y: 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. $\newline$ $x + 5 = -\left(\frac{2}{3}\right)y$
Multiply by $-\frac{3}{2}$ : Next, we multiply both sides of the equation by $-\frac{3}{2}$ to solve for $y$ . This will cancel out the $-\left(\frac{2}{3}\right)$ coefficient on the right side. $\newline$ $-\frac{3}{2} \times (x + 5) = y$
Simplify the equation: We distribute the $-\frac{3}{2}$ across the terms in the parentheses to simplify the equation. $\newline$ $y = -\frac{3}{2} \times x - \frac{15}{2}$

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