A function f is defined by `f(x)=x+(1)/(x)`. Consider the following.
(1) `(f(x))^(2)=f(x^(2))+2`
(2) `(f(x))^(3)=f(x^(3)) +3f(x)`
Which of the above is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to verify two statements involving the function \( f(x) = x + \frac{1}{x} \).

### Step 1: Verify the first statement
The first statement is:
\[ (f(x))^2 = f(x^2) + 2 \]

**Calculate \( (f(x))^2 \):**
\[ f(x) = x + \frac{1}{x} \]
Squaring \( f(x) \):
\[ (f(x))^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
Thus,
\[ (f(x))^2 = x^2 + 2 + \frac{1}{x^2} \]

**Calculate \( f(x^2) + 2 \):**
Now, we calculate \( f(x^2) \):
\[ f(x^2) = x^2 + \frac{1}{x^2} \]
Adding 2:
\[ f(x^2) + 2 = \left(x^2 + \frac{1}{x^2}\right) + 2 = x^2 + 2 + \frac{1}{x^2} \]

**Compare both sides:**
Now we see that:
\[ (f(x))^2 = x^2 + 2 + \frac{1}{x^2} = f(x^2) + 2 \]
Thus, the first statement is **true**.

### Step 2: Verify the second statement
The second statement is:
\[ (f(x))^3 = f(x^3) + 3f(x) \]

**Calculate \( (f(x))^3 \):**
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):
Let \( a = x \) and \( b = \frac{1}{x} \):
\[ (f(x))^3 = \left(x + \frac{1}{x}\right)^3 = x^3 + 3x\left(\frac{1}{x}\right)(x + \frac{1}{x}) + \frac{1}{x^3} \]
This simplifies to:
\[ x^3 + 3\left(x + \frac{1}{x}\right) + \frac{1}{x^3} = x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) \]

**Calculate \( f(x^3) + 3f(x) \):**
Now, we calculate \( f(x^3) \):
\[ f(x^3) = x^3 + \frac{1}{x^3} \]
Adding \( 3f(x) \):
\[ 3f(x) = 3\left(x + \frac{1}{x}\right) = 3x + \frac{3}{x} \]
Thus,
\[ f(x^3) + 3f(x) = \left(x^3 + \frac{1}{x^3}\right) + \left(3x + \frac{3}{x}\right) \]

**Compare both sides:**
Now we see that:
\[ (f(x))^3 = x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) \]
This matches with:
\[ f(x^3) + 3f(x) = x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) \]
Thus, the second statement is also **true**.

### Conclusion
Both statements (1) and (2) are correct.