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

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

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Full solution

Q. What is the inverse of the function\newlineg(x)=2x−1x+3?g−1(x)= \begin{array}{l} g(x)=\frac{2 x-1}{x+3} ? \\ g^{-1}(x)= \end{array}
  1. Replace g(x)g(x) with yy: To find the inverse of the function g(x)=2x−1x+3g(x) = \frac{2x - 1}{x + 3}, we need to switch the roles of xx and yy and then solve for yy. Let's start by replacing g(x)g(x) with yy:y=2x−1x+3y = \frac{2x - 1}{x + 3}
  2. Switch x and y: Now we switch x and y to find the inverse:\newlinex=2y−1y+3x = \frac{2y - 1}{y + 3}
  3. Multiply both sides by (y+3)(y + 3): Next, we solve for yy. To do this, we'll multiply both sides of the equation by (y+3)(y + 3) to eliminate the denominator:\newlinex(y+3)=2y−1x(y + 3) = 2y - 1
  4. Distribute xx on the left side: Distribute xx on the left side of the equation: xy+3x=2y−1xy + 3x = 2y - 1
  5. Move terms involving yy to the left side: To isolate yy, we need to get all the terms with yy on one side and the constant terms on the other side. Let's move the terms involving yy to the left side and the constant terms to the right side:\newlinexy−2y=−1−3xxy - 2y = -1 - 3x
  6. Factor out yy: Factor out yy from the left side of the equation:\newliney(x−2)=−1−3xy(x - 2) = -1 - 3x
  7. Divide both sides by (x−2)(x - 2): Now, divide both sides by (x−2)(x - 2) to solve for yy:y=−1−3xx−2y = \frac{-1 - 3x}{x - 2}
  8. Simplify the right side: We can simplify the right side of the equation by distributing the negative sign:\newliney=−1−3xx−2=−1x−2−3xx−2y = \frac{-1 - 3x}{x - 2} = \frac{-1}{x - 2} - \frac{3x}{x - 2}
  9. Inverse function found: The expression is already simplified, so we have found the inverse function: g−1(x)=−1x−2−3xx−2g^{-1}(x) = -\frac{1}{x - 2} - \frac{3x}{x - 2}

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