Questions: For f(x)=x^3+8, find each of the following. a. f(-1) b. f^(-1)(7). c. (f ∘ f^(-1))(476) a. f(-1)= b. f^(-1)(7)= c. (f ∘ f^(-1))(476)=

Solution Steps

To solve the given problems, we need to evaluate the function \( f(x) = x^3 + 8 \) and its inverse at specific points.

a. To find \( f(-1) \), substitute \( x = -1 \) into the function \( f(x) \).

b. To find \( f^{-1}(7) \), solve the equation \( 7 = x^3 + 8 \) for \( x \).

c. To find \( (f \circ f^{-1})(476) \), use the property that \( (f \circ f^{-1})(x) = x \).

To find \( f(-1) \), substitute \( x = -1 \) into the function \( f(x) = x^3 + 8 \): \[ f(-1) = (-1)^3 + 8 = -1 + 8 = 7 \]

To find \( f^{-1}(7) \), solve the equation \( 7 = x^3 + 8 \) for \( x \): \[ 7 = x^3 + 8 \implies x^3 = -1 \implies x = -1 \]

By the property of inverse functions, \( (f \circ f^{-1})(x) = x \): \[ (f \circ f^{-1})(476) = 476 \]

Final Answer