Questions: For f(x)=x^3-9, find each of the following. a. f(-3) b. f^(-1)(-36) c. (f ∘ f^(-1))(465) a. f(-3)= b. f^(-1)(-36)= c. (f ∘ f^(-1))(465)=

Solution Steps

a. To find $f(-3)$, substitute $-3$ into the function $f(x) = x^3 - 9$ and calculate the result.

b. To find $f^{-1}(-36)$, solve the equation $x^3 - 9 = -36$ for $x$. This involves finding the inverse of the function and then substituting $-36$ into it.

c. To find $(f \circ f^{-1})(465)$, use the property of functions and their inverses, which states that $(f \circ f^{-1})(x) = x$ for any $x$ in the domain of $f^{-1}$.

To find $f(-3)$, substitute $-3$ into the function $f(x) = x^3 - 9$:

$$f(-3) = (-3)^3 - 9 = -27 - 9 = -36$$

To find $f^{-1}(-36)$, solve the equation $x^3 - 9 = -36$ for $x$:

$$x^3 - 9 = -36 \implies x^3 = -36 + 9 = -27$$

Taking the cube root of both sides:

$$x = \sqrt[3]{-27} = -3$$

The composition of a function and its inverse, $(f \circ f^{-1})(x)$, is equal to $x$ for any $x$ in the domain of $f^{-1}$. Therefore:

$$(f \circ f^{-1})(465) = 465$$

Final Answer