Find the Inverse f(x) = square root of 25-x^2

Problem

ƒ(x)=√(,25−x^2)

Solution

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

y=√(,25−x^2)

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

x=√(,25−y^2)

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

x^2=25−y^2

4. Isolate the y^2 term by subtracting 25 from both sides and then multiplying by −1

y^2=25−x^2

5. Solve for y by taking the square root of both sides.

y=√(,25−x^2)

6. Restrict the domain. Since the original function ƒ(x)=√(,25−x^2) has a range of [0,5] the domain of the inverse function ƒ(x)^(−1) must be [0,5]

Final Answer

ƒ(x)^(−1)=√(,25−x^2),0≤x≤5

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