Let's solve the equation [tex]\( 3 \sin^{-1} (x) = \sin^{-1} (3x - 4x^3) \)[/tex] step by step.
### Step 1: Setting up the equation
We start with the equation:
[tex]\[ 3 \sin^{-1} (x) = \sin^{-1} (3x - 4x^3) \][/tex]
### Step 2: Apply the sine function to both sides
To remove the inverse sine (arcsin) function, we apply the sine function to both sides. However, we have to ensure that the arguments fall within the range [tex]\([-1, 1]\)[/tex] for the sine function to be valid.
[tex]\[ \sin \left( 3 \sin^{-1} (x) \right) = \sin \left( \sin^{-1} (3x - 4x^3) \right) \][/tex]
Since [tex]\( \sin (\sin^{-1} y) = y \)[/tex], the right-hand side simplifies to:
[tex]\[ \sin \left( 3 \sin^{-1} (x) \right) = 3x - 4x^3 \][/tex]
### Step 3: Use the triple angle formula for sine
The left-hand side involves taking the sine of three times an angle. We can use the sine triple angle formula:
[tex]\[ \sin (3\theta) = 3 \sin (\theta) - 4 \sin^3 (\theta) \][/tex]
In our case, [tex]\(\theta = \sin^{-1} (x)\)[/tex].
Thus,
[tex]\[ \sin \left( 3 \sin^{-1} (x) \right) = 3 \sin (\sin^{-1} (x)) - 4 \sin^3 (\sin^{-1} (x)) \][/tex]
Since [tex]\(\sin (\sin^{-1} (x)) = x\)[/tex],
[tex]\[ \sin \left( 3 \sin^{-1} (x) \right) = 3x - 4x^3 \][/tex]
### Step 4: Equating both sides
Now, we have:
[tex]\[ 3x - 4x^3 = 3x - 4x^3 \][/tex]
This simplifies to an identity, implying that the equation holds true for all [tex]\(x\)[/tex] within the domain of the original functions.
### Step 5: Check the domain restrictions
The main restriction comes from the domain of the [tex]\( \sin^{-1} \)[/tex] function, which is [tex]\([-1, 1]\)[/tex]. So, we also need to check the condition:
[tex]\[ 3x - 4x^3 \in [-1, 1] \][/tex]
This requires an analysis within the interval:
- Evaluate [tex]\(3x - 4x^3\)[/tex] over the interval [tex]\([-1, 1]\)[/tex].
We know:
- For [tex]\(x \in [-1, 1]\)[/tex], [tex]\(3x - 4x^3\)[/tex] lies in [tex]\([-1, 1]\)[/tex] since [tex]\(3x - 4x^3\)[/tex] is bounded within this range for [tex]\(x\)[/tex] in [tex]\([-1, 1]\)[/tex].
### Step 6: Final values for [tex]\(x\)[/tex]
Given the domain is respected within [tex]\([-1, 1]\)[/tex], we have:
[tex]\[ x \in [-1, 1] \][/tex]
Thus the solution is:
[tex]\[ \boxed{x \in [-1, 1]} \][/tex]
