If f: $R\rightarrow R$ be given by $f(x)=(3-x^{3})^{1/3}$, then fof (x) is

Q: If f: $R\rightarrow R$ be given by $f(x)=(3-x^{3})^{1/3}$, then fof (x) is

To solve the problem, we need to find $ f(f(x)) $ for the given function $ f(x) = (3 - x^3)^{1/3} $. 1. **Calculate $ f(f(x)) $:** Given $ f(x) = (3 - x^3)^{1/3} $, we substitute $ f(x) $ into itself: \[ f(f(x)) = f((3 - x^3)^{1/3}) \] Now, substitute $ (3 - x^3)^{1/3} $ into the function: \[ f((3 - x^3)^{1/3}) = \left(3 - \left((3 - x^3)^{1/3}\right)^3\right)^{1/3} \] Simplify the expression inside the function: \[ \left((3 - x^3)^{1/3}\right)^3 = 3 - x^3 \] Therefore, the expression becomes: \[ f((3 - x^3)^{1/3}) = \left(3 - (3 - x^3)\right)^{1/3} = \left(x^3\right)^{1/3} \] Simplifying further, we get: \[ (x^3)^{1/3} = x \] Thus, $ f(f(x)) = x $. 2. **Explanation of why Option 3 is correct:** The calculation above shows that when we apply the function $ f $ to itself, the result simplifies to $ x $. This matches Option 3, which is $ x $. 3. **Explanation of why other options are incorrect:** - **Option 1: $ x^{1/3} $** This option suggests that $ f(f(x)) = x^{1/3} $. However, our calculation shows that $ f(f(x)) = x $, not $ x^{1/3} $. - **Option 2: $ x^3 $** This option suggests that $ f(f(x)) = x^3 $. Again, our calculation shows that $ f(f(x)) = x $, not $ x^3 $. - **Option 4: $ (3-x^3) $** This option suggests that $ f(f(x)) = 3 - x^3 $. Our calculation clearly shows that $ f(f(x)) = x $, not $ 3 - x^3 $. In conclusion, the correct answer is Option 3, $ x $, as demonstrated by the step-by-step calculation of $ f(f(x)) $.

Question

Accepted Answer

To solve the problem, we need to find $ f(f(x)) $ for the given function $ f(x) = (3 - x^3)^{1/3} $.

1. **Calculate $ f(f(x)) $:**

Given $ f(x) = (3 - x^3)^{1/3} $, we substitute $ f(x) $ into itself:

$$
   f(f(x)) = f((3 - x^3)^{1/3})
   $$

Now, substitute $ (3 - x^3)^{1/3} $ into the function:

$$
   f((3 - x^3)^{1/3}) = \left(3 - \left((3 - x^3)^{1/3}\right)^3\right)^{1/3}
   $$

Simplify the expression inside the function:

$$
   \left((3 - x^3)^{1/3}\right)^3 = 3 - x^3
   $$

Therefore, the expression becomes:

$$
   f((3 - x^3)^{1/3}) = \left(3 - (3 - x^3)\right)^{1/3} = \left(x^3\right)^{1/3}
   $$

Simplifying further, we get:

$$
   (x^3)^{1/3} = x
   $$

Thus, $ f(f(x)) = x $.

2. **Explanation of why Option 3 is correct:**

The calculation above shows that when we apply the function $ f $ to itself, the result simplifies to $ x $. This matches Option 3, which is $ x $.

3. **Explanation of why other options are incorrect:**

- **Option 1: $ x^{1/3} $**

This option suggests that $ f(f(x)) = x^{1/3} $. However, our calculation shows that $ f(f(x)) = x $, not $ x^{1/3} $.

- **Option 2: $ x^3 $**

This option suggests that $ f(f(x)) = x^3 $. Again, our calculation shows that $ f(f(x)) = x $, not $ x^3 $.

- **Option 4: $ (3-x^3) $**

This option suggests that $ f(f(x)) = 3 - x^3 $. Our calculation clearly shows that $ f(f(x)) = x $, not $ 3 - x^3 $.

In conclusion, the correct answer is Option 3, $ x $, as demonstrated by the step-by-step calculation of $ f(f(x)) $.