Question

If \( f: \mathbb{R} \to \mathbb{R} \), is given by \( f(x) = (3 - x^3)^{1/3} \), then \( f \circ f(x) \) is equal to :

• A. \( x^{1/3} \)
• B. \( x^3 \)
• C. \( x \)
• D. \( (3 - x^3) \)

Hint:

When dealing with function composition, work from the inside out. First, evaluate the inner function, then use its result as the input for the outer function. For functions that seem to "undo" themselves, like this one, they are their own inverses. For any function \( f \) that is its own inverse, \( f(f(x)) = x \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The notation \( fof(x) \) represents the composition of the function \( f \) with itself, which is also written as \( f(f(x)) \). To find this, we substitute the entire expression for \( f(x) \) into the variable \( x \) within the function \( f \) again.
Step 2: Key Formula or Approach:
Given \( f(x) = (3 - x^3)^{1/3} \).
We need to calculate \( f(f(x)) \).
Step 3: Detailed Explanation or Calculation:
Start with the definition of \( fof(x) \):
\[ fof(x) = f(f(x)) \] Substitute the expression for \( f(x) \):
\[ fof(x) = f\left((3 - x^3)^{1/3}\right) \] Now, apply the function \( f \) to this new input. This means we replace \( x \) in the expression \( (3 - x^3)^{1/3} \) with the input \( (3 - x^3)^{1/3} \).
\[ fof(x) = \left(3 - \left[(3 - x^3)^{1/3}\right]^3\right)^{1/3} \] The cube and the cube root cancel each other out:
\[ fof(x) = \left(3 - (3 - x^3)\right)^{1/3} \] Simplify the expression inside the parenthesis:
\[ fof(x) = (3 - 3 + x^3)^{1/3} \] \[ fof(x) = (x^3)^{1/3} \] \[ fof(x) = x \] Step 4: Final Answer:
The value of \( fof(x) \) is \( x \).