Questions: Verify that the equation is correct. f(f^-1(x))=f(x-8) and f^-1(f(x))=f^-1(x+8 Substitute. =square=square Simplify.

Solution Steps

To verify the given equations, we need to check if the compositions of the functions \( f \) and \( f^{-1} \) hold true. Specifically, we need to verify if \( f(f^{-1}(x)) = f(x-8) \) and \( f^{-1}(f(x)) = f^{-1}(x+8) \).

We start by defining the functions \( f(x) \) and \( f^{-1}(x) \): \[ f(x) = x + 8 \] \[ f^{-1}(x) = x - 8 \]

We need to verify if \( f(f^{-1}(x)) = f(x-8) \).

1. Compute \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f(x - 8) = (x - 8) + 8 = x \]

2. Compute \( f(x-8) \): \[ f(x-8) = (x - 8) + 8 = x \]

Since both sides are equal: \[ f(f^{-1}(x)) = f(x-8) = x \]

We need to verify if \( f^{-1}(f(x)) = f^{-1}(x+8) \).

1. Compute \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}(x + 8) = (x + 8) - 8 = x \]

2. Compute \( f^{-1}(x+8) \): \[ f^{-1}(x+8) = (x + 8) - 8 = x \]

Since both sides are equal: \[ f^{-1}(f(x)) = f^{-1}(x+8) = x \]

Final Answer