Find the Inverse f(x)=x/(x-5)

Problem

ƒ(x)=x/(x−5)

Solution

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

y=x/(x−5)

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

x=y/(y−5)

3. Multiply both sides by the denominator (y−5) to clear the fraction.

x*(y−5)=y

4. Distribute x into the parentheses.

x*y−5*x=y

5. Rearrange the equation to group all terms containing y on one side and terms without y on the other.

x*y−y=5*x

6. Factor out the common variable y from the left side.

y*(x−1)=5*x

7. Isolate y by dividing both sides by (x−1)

y=(5*x)/(x−1)

8. Rewrite y using the inverse function notation ƒ(x)(−1)

ƒ(x)(−1)=(5*x)/(x−1)

Final Answer

ƒ(x)(−1)=(5*x)/(x−1)

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