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

Problem

ƒ(x)=4/(1+4)

Solution

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

y=4/(1+4)

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

x=4/(1+4)

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

x*(1+4)=4

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

x+x⋅4=4

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

x=4−x⋅4

6. Factor out 4 from the right side.

x=4*(1−x)

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

x/(1−x)=4

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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