Find the Inverse f(x) = square root of 16-x^2

Problem

ƒ(x)=√(,16−x^2)

Solution

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

y=√(,16−x^2)

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

x=√(,16−y^2)

3. Square both sides of the equation to eliminate the radical.

x^2=16−y^2

4. Isolate the y^2 term by subtracting 16 from both sides and multiplying by −1

y^2=16−x^2

5. Solve for y by taking the square root of both sides. Note that for the function to be invertible, we typically restrict the domain. For the original function ƒ(x)=√(,16−x^2) the range is [0,4] which becomes the domain of the inverse.

y=√(,16−x^2)

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

ƒ(x)^(−1)=√(,16−x^2)

Final Answer

ƒ(x)^(−1)=√(,16−x^2)

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