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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\newlineg(x)=−x−2x+4?g−1(x)=□ \begin{array}{l} g(x)=\frac{-x-2}{x+4} ? \\ g^{-1}(x)=\square \end{array}

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Full solution

Q. What is the inverse of the function\newlineg(x)=−x−2x+4?g−1(x)=□ \begin{array}{l} g(x)=\frac{-x-2}{x+4} ? \\ g^{-1}(x)=\square \end{array}
  1. Switching roles and setting up the equation: To find the inverse of the function g(x)g(x), we need to switch the roles of xx and g(x)g(x) and then solve for the new xx.
    Let y=g(x)y = g(x), so we have:
    y=−x−2x+4y = \frac{-x - 2}{x + 4}
    Now we switch xx and yy:
    x=−y−2y+4x = \frac{-y - 2}{y + 4}
  2. Multiplying both sides to eliminate the fraction: Next, we need to solve for yy. To do this, we'll multiply both sides of the equation by (y+4)(y + 4) to get rid of the fraction:\newlinex(y+4)=−y−2x(y + 4) = -y - 2\newlinexy+4x=−y−2xy + 4x = -y - 2
  3. Rearranging the equation: Now, we'll move all terms involving yy to one side of the equation and the constant terms to the other side: xy+y=−4x−2xy + y = -4x - 2 y(x+1)=−4x−2y(x + 1) = -4x - 2
  4. Isolating yy to find the inverse function: To isolate yy, we divide both sides of the equation by (x+1)(x + 1):y=−4x−2x+1y = \frac{-4x - 2}{x + 1}This is the inverse function of g(x)g(x).

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