f(x) = `(x^n + (pi/3)^n)/(x^(n-1) + (pi/3)^(n-1))`,(n is an even number, then which of the following is correct

Options

A. If `f ,[(pi)/(3), oo)to [(pi)/(3), oo),` then function is inbvertible
B. `f (x) = f(-x)|` has infinite number of solutions
C. `f (x)= |f (x)|` has infinite nouber of solution
D. `f (x)` is one-one function for all `x in R`

Answer

To solve the problem, we need to analyze the function given:

\[ f(x) = \frac{x^n + \left(\frac{\pi}{3}\right)^n}{x^{n-1} + \left(\frac{\pi}{3}\right)^{n-1}} \]

where \( n \) is an even number.

### Step 1: Simplify the Function

We can simplify the function by dividing the numerator and the denominator by \( x^{n-1} \):

\[ f(x) = \frac{\frac{x^n}{x^{n-1}} + \frac{\left(\frac{\pi}{3}\right)^n}{x^{n-1}}}{\frac{x^{n-1}}{x^{n-1}} + \frac{\left(\frac{\pi}{3}\right)^{n-1}}{x^{n-1}}} \]

This simplifies to:

\[ f(x) = \frac{x + \frac{\left(\frac{\pi}{3}\right)^n}{x^{n-1}}}{1 + \frac{\left(\frac{\pi}{3}\right)^{n-1}}{x^{n-1}}} \]

### Step 2: Take the Limit as \( x \to \infty \)

Next, we will evaluate the limit of \( f(x) \) as \( x \) approaches infinity:

\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x + 0}{1 + 0} = \lim_{x \to \infty} x = \infty \]

### Step 3: Analyze the Function's Behavior

Since \( n \) is even, the function \( f(x) \) is continuous and differentiable for \( x > \frac{\pi}{3} \). We need to check if the function is one-to-one (invertible).

### Step 4: Check for Monotonicity

To check if \( f(x) \) is one-to-one, we can find its derivative:

\[ f'(x) = \frac{(1)(x^{n-1} + \left(\frac{\pi}{3}\right)^{n-1}) - (x^n + \left(\frac{\pi}{3}\right)^n)(n-1)x^{n-2}}{(x^{n-1} + \left(\frac{\pi}{3}\right)^{n-1})^2} \]

Since \( n \) is even, \( f'(x) \) will be positive for \( x > \frac{\pi}{3} \), indicating that \( f(x) \) is increasing.

### Step 5: Conclusion on Invertibility

Since \( f(x) \) is continuous, differentiable, and strictly increasing for \( x > \frac{\pi}{3} \), it is a one-to-one function. Thus, it is invertible.

### Final Answer

The correct option is that the function is invertible.

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