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)=x3−9 and calculate the result.
b. To find f−1(−36), solve the equation x3−9=−36 for x. This involves finding the inverse of the function and then substituting −36 into it.
c. To find (f∘f−1)(465), use the property of functions and their inverses, which states that (f∘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)=x3−9:
f(−3)=(−3)3−9=−27−9=−36
Step 2: Solve for f−1(−36)
To find f−1(−36), solve the equation x3−9=−36 for x:
x3−9=−36⟹x3=−36+9=−27
Taking the cube root of both sides:
x=3−27=−3
Step 3: Evaluate (f∘f−1)(465)
The composition of a function and its inverse, (f∘f−1)(x), is equal to x for any x in the domain of f−1. Therefore: