To solve [tex]\( f(x) = \sin \sqrt{6x^4 - 8x - 3} \)[/tex], we need to evaluate the given function step-by-step. Let's break down this complex function into more manageable parts.
1. Examine the Inner Expression:
- The expression inside the sine function is [tex]\( \sqrt{6x^4 - 8x - 3} \)[/tex].
2. Simplify the Inner Expression:
- Calculate the term [tex]\( 6x^4 \)[/tex]. This means raising [tex]\( x \)[/tex] to the fourth power and multiplying the result by 6.
- Calculate the term [tex]\( -8x \)[/tex]. This means multiplying [tex]\( x \)[/tex] by -8.
- Subtract 3 from the sum of these two terms.
3. Form the Argument of the Sine Function:
- Take the square root of the expression [tex]\( 6x^4 - 8x - 3 \)[/tex]. This is done by power [tex]\( \frac{1}{2} \)[/tex].
4. Apply the Sine Function:
- Finally, apply the sine function to the result obtained from step 3.
Thus, for a given value of [tex]\( x \)[/tex], the step-by-step computation would look like:
- Compute [tex]\( 6x^4 \)[/tex].
- Compute [tex]\( -8x \)[/tex].
- Sum these results and subtract 3.
- Take the square root of the outcome.
- Apply the sine function to this square root.
Therefore, the function [tex]\( f(x) \)[/tex] can be expressed as:
[tex]\[ f(x) = \sin \left( \sqrt{6x^4 - 8x - 3} \right) \][/tex]
This is the detailed step-by-step solution for the function [tex]\( f(x) \)[/tex].
