8. Verify that the two given functions are inverses of each other (i) \( f(x)=\sqrt{2 x-4} \) for \( x \geq 0 \) and \( g(x)=\frac{x^{2}+4}{2} \) (ii) \( f(x)=x^{2}-4 \) for \( x \geq 0 \) and \( g(x)=\sqrt{x+4} \) for \( x \geq-4 \)

To verify that two functions are inverses of each other, we need to show that \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the respective domains. For the first pair: 1. Start with \( g(x) = \frac{x^2 + 4}{2} \). Now compute \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x^2 + 4}{2}\right) = \sqrt{2\left(\frac{x^2 + 4}{2}\right) - 4} = \sqrt{x^2} = |x| = x \quad (x \geq 0) \] 2. Now compute \( g(f(x)) \): \[ g(f(x)) = g(\sqrt{2x - 4}) = \frac{(\sqrt{2x - 4})^2 + 4}{2} = \frac{2x - 4 + 4}{2} = x \quad (x \geq 0) \] They are indeed inverses. For the second pair: 1. Start with \( g(x) = \sqrt{x + 4} \). Now compute \( f(g(x)) \): \[ f(g(x)) = f(\sqrt{x + 4}) = (\sqrt{x + 4})^2 - 4 = x + 4 - 4 = x \quad (x \geq -4) \] 2. Now compute \( g(f(x)) \): \[ g(f(x)) = g(x^2 - 4) = \sqrt{(x^2 - 4) + 4} = \sqrt{x^2} = |x| = x \quad (x \geq 0) \] These functions are also inverses. In conclusion, both pairs of functions are inverses of each other!