For the following exercises, find the inverse of the function on the given domain 5. \( f(x)=(x-4)^{2},[4, \infty) \) 6. \( f(x)=(x+2)^{2},[-2, \infty) \) 7. \( f(x)=(x+1)^{2}-3,[-1, \infty) \) 8. \( f(x)=2-\sqrt{3+x} \) 9. \( f(x)=3 x^{2}+5,(-\infty, 0] \) 10. \( f(x)=12-x^{2},[0, \infty) \) 11. \( f(x)=9-x^{2},[0, \infty) \)

To find the inverse of the given functions on the specified domains, we will follow these steps: 1. Replace \( f(x) \) with \( y \). 2. Solve for \( x \) in terms of \( y \). 3. Swap \( x \) and \( y \) to express the inverse function. Let's go through each function one by one. ### 5. \( f(x) = (x-4)^{2}, [4, \infty) \) 1. Set \( y = (x-4)^{2} \). 2. Solve for \( x \): \[ \sqrt{y} = x - 4 \quad \text{(since \( x \geq 4 \), we take the positive root)} \] \[ x = \sqrt{y} + 4 \] 3. Swap \( x \) and \( y \): \[ f^{-1}(x) = \sqrt{x} + 4, \quad [0, \infty) \] ### 6. \( f(x) = (x+2)^{2}, [-2, \infty) \) 1. Set \( y = (x+2)^{2} \). 2. Solve for \( x \): \[ \sqrt{y} = x + 2 \quad \text{(since \( x \geq -2 \), we take the positive root)} \] \[ x = \sqrt{y} - 2 \] 3. Swap \( x \) and \( y \): \[ f^{-1}(x) = \sqrt{x} - 2, \quad [0, \infty) \] ### 7. \( f(x) = (x+1)^{2} - 3, [-1, \infty) \) 1. Set \( y = (x+1)^{2} - 3 \). 2. Solve for \( x \): \[ y + 3 = (x+1)^{2} \] \[ \sqrt{y + 3} = x + 1 \quad \text{(since \( x \geq -1 \), we take the positive root)} \] \[ x = \sqrt{y + 3} - 1 \] 3. Swap \( x \) and \( y \): \[ f^{-1}(x) = \sqrt{x + 3} - 1, \quad [3, \infty) \] ### 8. \( f(x) = 2 - \sqrt{3+x} \) 1. Set \( y = 2 - \sqrt{3+x} \). 2. Solve for \( x \): \[ \sqrt{3+x} = 2 - y \] \[ 3 + x = (2 - y)^{2} \] \[ x = (2 - y)^{2} - 3 \] 3. Swap \( x \) and \( y \): \[ f^{-1}(x) = (2 - x)^{2} - 3, \quad (-\infty, 2] \] ### 9. \( f(x) = 3x^{2} + 5, (-\infty, 0] \) 1. Set \( y = 3x^{2} + 5 \). 2. Solve for \( x \): \[ y - 5 = 3x^{2} \] \[ x^{2} = \frac{y - 5}{3} \] \[ x = -\sqrt{\frac{y - 5}{3}} \quad \text{(since \( x \leq 0 \))} \] 3. Swap \( x \) and \( y \): \[ f^{-1}(x) = -\sqrt{\frac{x - 5}{3}}, \quad [5, \infty) \] ### 10. \( f(x) = 12 - x^{2}, [0, \infty) \) 1. Set \( y = 12 - x^{2} \). 2. Solve for \( x \): \[ x^{2} = 12 - y \] \[ x = \sqrt{12 - y} \quad \text{(since \( x \geq 0 \))} \] 3. Swap \( x \) and \( y