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Find the inverse of each function. f(x)=(-2-3x)/(2)

Find the inverse of each function.\newline f(x)=−2−3x2 f(x)=\frac{-2-3 x}{2}

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Full solution

Q. Find the inverse of each function.\newline f(x)=−2−3x2 f(x)=\frac{-2-3 x}{2}
  1. Switch Roles and Solve: To find the inverse of the function f(x)=−2−3x2f(x) = \frac{-2 - 3x}{2}, we need to switch the roles of xx and f(x)f(x) and then solve for the new xx. Let y=f(x)y = f(x), so we have y=−2−3x2y = \frac{-2 - 3x}{2}. Now we switch xx and yy to get x=−2−3y2x = \frac{-2 - 3y}{2}.
  2. Multiply and Isolate: Next, we need to solve for yy. Start by multiplying both sides of the equation by 22 to get rid of the denominator.2×x=−2−3y2 \times x = -2 - 3y
  3. Divide and Solve: Now, we isolate the term containing yy by adding 22 to both sides of the equation.2x+2=−3y2x + 2 = -3y
  4. Final Inverse Function: To solve for yy, we divide both sides of the equation by −3-3.\newliney=2x+2−3y = \frac{2x + 2}{-3}
  5. Final Inverse Function: To solve for yy, we divide both sides of the equation by −3-3.y=2x+2−3y = \frac{2x + 2}{-3}We have found the inverse function. The inverse function of f(x)=−2−3x2f(x) = \frac{-2 - 3x}{2} is f−1(x)=2x+2−3f^{-1}(x) = \frac{2x + 2}{-3}.

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