If `x in [-1,-1/2]` then express the function `f(x)=sin^(-1) (3x-4x^(3))+cos^(-1) (4x^(3)-3x)` in the form of a `cos^(-1) x+bpi`, where a and b are rational numbers.

Answer

To express the function \( f(x) = \sin^{-1}(3x - 4x^3) + \cos^{-1}(4x^3 - 3x) \) in the form \( \cos^{-1}(x) + b\pi \), we will follow these steps:

### Step 1: Recognize the Identity
We know that:
\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]
for any \( y \) in the range \([-1, 1]\).

### Step 2: Apply the Identity
Let \( y = 3x - 4x^3 \). Then we can rewrite \( f(x) \):
\[ f(x) = \sin^{-1}(y) + \cos^{-1}(4x^3 - 3x) \]
We need to check if \( 4x^3 - 3x = y \).

### Step 3: Check the Relationship
Calculate \( 4x^3 - 3x \):
\[ 4x^3 - 3x = 4(3x - 4x^3) - 3(3x - 4x^3) \]
This simplifies to:
\[ 4y - 3y = y \]
Thus, \( 4x^3 - 3x = 3x - 4x^3 \).

### Step 4: Substitute Back
Since \( 4x^3 - 3x = y \), we can rewrite \( f(x) \):
\[ f(x) = \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]

### Step 5: Determine the Form
Now, we need to express \( f(x) \) in the form \( \cos^{-1}(x) + b\pi \). Since \( f(x) = \frac{\pi}{2} \), we can express it as:
\[ f(x) = \cos^{-1}(0) + \frac{1}{2}\pi \]
Here, \( a = 0 \) and \( b = \frac{1}{2} \).

### Conclusion
Thus, the function can be expressed as:
\[ f(x) = \cos^{-1}(0) + \frac{1}{2}\pi \]