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)=

Transcript text: For $f(x)=x^{3}-9$, find each of the following. a. $f(-3)$ b. $f^{-1}(-36)$ c. $\left(f \circ f^{-1}\right)(465)$ a. $f(-3)=$ $\square$ b. $f^{-1}(-36)=$ $\square$ c. $\left(f \circ f^{-1}\right)(465)=$ $\square$

Solution

Solution Steps

Solution Approach

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} $.

Step 1: Calculate $ f(-3) $

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

\[ f(-3) = (-3)^3 - 9 = -27 - 9 = -36 \]

Step 2: Solve for $ f^{-1}(-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:

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

Step 3: Evaluate $ (f \circ f^{-1})(465) $

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:

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