Find the Inverse f(x)=e^(2x-1)

Problem

ƒ(x)=e(2*x−1)

Solution

1. Replace ƒ(x) with y to set up the equation for the inverse.

y=e(2*x−1)

2. Swap the variables x and y to begin solving for the inverse function.

x=e(2*y−1)

3. Apply the natural logarithm to both sides of the equation to isolate the exponent.

ln(x)=ln(e(2*y−1))

4. Simplify the right side using the property ln(eu)=u

ln(x)=2*y−1

5. Isolate the term containing y by adding 1 to both sides.

ln(x)+1=2*y

6. Solve for y by dividing both sides by 2

y=(ln(x)+1)/2

7. Replace y with ƒ(x)(−1) to express the final inverse function.

ƒ(x)(−1)=(ln(x)+1)/2

Final Answer

ƒ(x)(−1)=(ln(x)+1)/2

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