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)=

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. $\left(f \circ f^{-1}\right)(465)=$ $\square$
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Solution

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Solution Steps

Solution Approach

a. To find f(3) f(-3) , substitute 3-3 into the function f(x)=x39 f(x) = x^3 - 9 and calculate the result.

b. To find f1(36) f^{-1}(-36) , solve the equation x39=36 x^3 - 9 = -36 for x x . This involves finding the inverse of the function and then substituting 36-36 into it.

c. To find (ff1)(465) (f \circ f^{-1})(465) , use the property of functions and their inverses, which states that (ff1)(x)=x (f \circ f^{-1})(x) = x for any x x in the domain of f1 f^{-1} .

Step 1: Calculate f(3) f(-3)

To find f(3) f(-3) , substitute 3-3 into the function f(x)=x39 f(x) = x^3 - 9 :

f(3)=(3)39=279=36 f(-3) = (-3)^3 - 9 = -27 - 9 = -36

Step 2: Solve for f1(36) f^{-1}(-36)

To find f1(36) f^{-1}(-36) , solve the equation x39=36 x^3 - 9 = -36 for x x :

x39=36    x3=36+9=27 x^3 - 9 = -36 \implies x^3 = -36 + 9 = -27

Taking the cube root of both sides:

x=273=3 x = \sqrt[3]{-27} = -3

Step 3: Evaluate (ff1)(465) (f \circ f^{-1})(465)

The composition of a function and its inverse, (ff1)(x) (f \circ f^{-1})(x) , is equal to x x for any x x in the domain of f1 f^{-1} . Therefore:

(ff1)(465)=465 (f \circ f^{-1})(465) = 465

Final Answer

  • f(3)=36 f(-3) = \boxed{-36}
  • f1(36)=3 f^{-1}(-36) = \boxed{-3}
  • (ff1)(465)=465 (f \circ f^{-1})(465) = \boxed{465}
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