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[15] Let f(x)=ln(4-x)+1. Please find f^(-1)(x).

88. [1515] Let f(x)=ln⁡(4−x)+1 f(x)=\ln (4-x)+1 . Please find f−1(x) f^{-1}(x) .

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Q. 88. [1515] Let f(x)=ln⁡(4−x)+1 f(x)=\ln (4-x)+1 . Please find f−1(x) f^{-1}(x) .
  1. Let y=f(x)y = f(x): Let y=f(x)=ln⁡(4−x)+1y = f(x) = \ln(4-x) + 1. To find the inverse, we need to solve for xx in terms of yy.
  2. Subtract 11: Subtract 11 from both sides: y−1=ln⁡(4−x)y - 1 = \ln(4-x).
  3. Raise ee to power: Raise ee to the power of both sides to get rid of the natural log: e(y−1)=eln⁡(4−x)e^{(y-1)} = e^{\ln(4-x)}.
  4. Simplify right side: Simplify the right side using the property eln⁡(a)=ae^{\ln(a)} = a: ey−1=4−xe^{y-1} = 4-x.
  5. Solve for x: Solve for x: x=4−e(y−1)x = 4 - e^{(y-1)}.
  6. Replace yy with xx: Replace yy with xx since we're finding the inverse: f−1(x)=4−e(x−1)f^{-1}(x) = 4 - e^{(x-1)}.

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