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$What is the inverse of the function {:[g(x)=(-x-2)/(x+4)?],[g^(-1)(x)=◻]:}$

What is the inverse of the function $\newline$ $g(x) = \frac{-x - 2}{x + 4}$ ? $\newline$ $g^{-1}(x) = \square$

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Find the vertex of the transformed function

Full solution

Q. What is the inverse of the function

\newline

g(x) = \frac{-x - 2}{x + 4}

?

\newline

g^{-1}(x) = \square

Replace with y: To find the inverse of the function $g(x) = \frac{-x-2}{x+4}$ , we need to switch the roles of $x$ and $y$ , and then solve for $y$ . Let's start by replacing $g(x)$ with $y$ : $y = \frac{-x-2}{x+4}$
Switch x and y: Now, switch x and y to find the inverse: $x = \frac{-y-2}{y+4}$
Cross-multiply: Next, we need to solve for $y$ . To do this, we'll start by cross-multiplying to get rid of the fraction: $\newline$ $x(y + 4) = -y - 2$
Distribute $x$ : Distribute the $x$ on the left side of the equation: $\newline$ $xy + 4x = -y - 2$
Combine like terms: Now, we want to get all the terms with $y$ on one side and the constant terms on the other side. Let's add $y$ to both sides and add $2$ to both sides: $xy + y + 4x + 2 = 0$
Isolate $y$ : Combine like terms by factoring $y$ out of the terms on the left side: $y(x + 1) + 4x + 2 = 0$
Divide both sides: Now, isolate the term with $y$ by subtracting $4x$ and $2$ from both sides: $\newline$ $y(x + 1) = -4x - 2$
Find inverse: Finally, divide both sides by $(x + 1)$ to solve for $y$ : $y = \frac{-4x - 2}{x + 1}$
Find inverse: Finally, divide both sides by $(x + 1)$ to solve for $y$ : $y = \frac{-4x - 2}{x + 1}$ We have found the inverse function. The inverse of $g(x)$ is: $g^{-1}(x) = \frac{-4x - 2}{x + 1}$

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