Find the Inverse f(x)=(4^x)/(1+4^x)

Problem

ƒ(x)=4^x/(1+4^x)

Solution

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

y=4^x/(1+4^x)

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

x=4^y/(1+4^y)

3. Multiply both sides by the denominator (1+4^y) to clear the fraction.

x*(1+4^y)=4^y

4. Distribute x on the left side of the equation.

x+x⋅4^y=4^y

5. Isolate the terms containing 4^y on one side by subtracting x⋅4^y from both sides.

x=4^y−x⋅4^y

6. Factor out 4^y from the right side.

x=4^y*(1−x)

7. Divide both sides by (1−x) to isolate the exponential term.

x/(1−x)=4^y

8. Apply the logarithm with base 4 to both sides to solve for y

(log_4)(x/(1−x))=y

9. Replace y with ƒ(x)^(−1) to state the final inverse function.

ƒ(x)^(−1)=(log_4)(x/(1−x))

Final Answer

ƒ(x)^(−1)=(log_4)(x/(1−x))

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