Calc 2 problem with inverse

Question

Let f(x) = sqrt(16-x^2),    0 ≤ x ≤ 4.

Find f ⁻¹

(for the given range)

Answer 1

To find the inverse of a function, you can use the following 4 step process,

1. Replace f(x) with y
2. Swap every x with every y in the equation
3. Solve the equation for y
4. Replace y with f^-1(x)

For this problem,

f(x) = √(16-x²)

y = √(16-x²) [step 1]

x = √(16-y²) [step 2]

x² = 16 - y²

y² = 16 - x²

y = √(16-x²) [step 3]

f^-1(x) = √(16-x²) [step 4]

So the inverse of the function is f^-1(x) = √(16-x²)

Note that in this case, the function is the inverse of itself.

To find the domain of the inverse, we only have to realize the the domain of the inverse is the range of the original function. So, we can plug in the end points of the domain of the function into f(x) to find the domain of the inverse.

f(x) = √(16-x²)

f(0)=√(16-0²) = √(16) = 4

f(4) = √(16-4²) = √(0) = 0

Therefore the domain of the inverse is between 0 and 4, i.e. 0 ≤ x ≤ 4

Answer 2

y = √ (16-x² )

Solve for x.

y² = 16 - x²

x² = 16 - y²

x = √ (16-y²)

Switch x, y

y = √ (16-x²)

This is the required inverse function.

By definition f ( f^-1 (x)) = x

Lets check f ( f^-1 (x)) = f (√ 16-x² ) = √ 16 - (√16-x²)² = √ 16 - (16-x²) = √x² = x as required