Kar Caraway

To prove that the inverse of \( f(x) = 1 + \sqrt{x} \) is \( f^{-1}(x) = (x - 1)^2 \), we verify that both compositions \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) simplify to \( x \), considering domain restrictions. 1. **Compose \( f(f^{-1}(x)) \):** \[ f(f^{-1}(x)) = f\left((x - 1)^2\right) = 1 + \sqrt{(x - 1)^2}. \] Since \( f^{-1}(x) \) has domain \( x \geq 1 \), \( x - 1 \geq 0 \). Thus: \[ \sqrt{(x - 1)^2} = |x - 1| = x - 1 \quad \text{(as \( x \geq 1 \))}. \] Therefore: \[ f(f^{-1}(x)) = 1 + (x - 1) = x. \] 2. **Compose \( f^{-1}(f(x)) \):** \[ f^{-1}(f(x)) = f^{-1}\left(1 + \sqrt{x}\right) = \left(1 + \sqrt{x} - 1\right)^2. \] Simplify: \[ \left(\sqrt{x}\right)^2 = x \quad \text{(valid for \( x \geq 0 \), the domain of \( f \))}. \] 3. **Deriving the Inverse Algebraically:** - Start with \( y = 1 + \sqrt{x} \). - Swap \( x \) and \( y \): \( x = 1 + \sqrt{y} \). - Solve for \( y \): \[ x - 1 = \sqrt{y} \implies y = (x - 1)^2. \] - Domain of \( f^{-1}(x) \): \( x \geq 1 \), matching the range of \( f \). **Conclusion:** With domain restrictions (\( f \) for \( x \geq 0 \), \( f^{-1} \) for \( x \geq 1 \)), the functions satisfy \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Thus, the inverse of \( f(x) = 1 + \sqrt{x} \) is indeed \( f^{-1}(x) = (x - 1)^2 \). \(\boxed{f^{-1}(x) = (x - 1)^2}\)

*Kar feels a bit of relief as he watches the man sleep, he's still a bit in shock but he can't deny the fact that he's attracted to this man. He slowly reaches over and gently brushes his fingers across the man's cheek*

*Kar stares at the man's lips and the urge to kiss him is overwhelming, he hesitates for a moment before leaning in slowly. His lips are inches from the man's when he pulls away and shakes his head* No, I can't do this, not while he's sleeping.

*He turns and looks at the sleeping woman, her beauty was breathtaking, her skin soft and supple. Her hair fell down in soft waves and her curves were perfect. Kar couldn't help but notice the generous amount of cleavage and the curve of her hips. She was absolutely gorgeous and Kar felt a surge of desire as he looked at her. He then noticed the ring on her finger, a sign that she was already taken. Kar's expression turned cold and he looked away, he needed answers and this woman could give them to him.*