Narišimo funkcijo \( f(x)=2 \sin \left(3 x-\frac{3 \pi}{4}\right)-1 \) in

Function by following steps: - step0: Determine the period: \(f\left(x\right)=2\sin\left(3x+\frac{5\pi }{4}\right)-1\) - step1: The period of the function is \(\frac{2\pi }{\left|3\right|}:\) \(\frac{2\pi }{\left|3\right|}\) - step2: Calculate: \(\frac{2\pi }{3}\) It seems there was an issue retrieving the results for the analysis of the function \( f(x) = 2 \sin \left(3 x - \frac{3 \pi}{4}\right) - 1 \). However, I can provide some insights based on the properties of sine functions and the transformations applied. ### Function Analysis 1. **Domain**: The domain of the sine function is all real numbers. Therefore, the domain of \( f(x) \) is: \[ \text{Domain: } (-\infty, \infty) \] 2. **Range**: The sine function oscillates between -1 and 1. The transformation \( 2 \sin \left(3 x - \frac{3 \pi}{4}\right) \) scales this range to \([-2, 2]\). After subtracting 1, the range becomes: \[ \text{Range: } [-3, 1] \] 3. **Critical Points**: To find the critical points, we need to take the derivative of \( f(x) \) and set it to zero. The derivative is: \[ f'(x) = 2 \cdot 3 \cos \left(3 x - \frac{3 \pi}{4}\right) = 6 \cos \left(3 x - \frac{3 \pi}{4}\right) \] Setting \( f'(x) = 0 \): \[ 6 \cos \left(3 x - \frac{3 \pi}{4}\right) = 0 \implies \cos \left(3 x - \frac{3 \pi}{4}\right) = 0 \] The cosine function is zero at odd multiples of \( \frac{\pi}{2} \): \[ 3 x - \frac{3 \pi}{4} = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z}) \] Solving for \( x \): \[ 3 x = \frac{3 \pi}{4} + \frac{\pi}{2} + k\pi \implies x = \frac{3 \pi}{12} + \frac{2\pi}{12} + \frac{k\pi}{3} = \frac{5\pi}{12} + \frac{k\pi}{3} \] 4. **Inflection Points**: Inflection points occur where the second derivative changes sign. We would need to compute the second derivative and analyze it, but this requires more detailed calculations. 5. **Intervals of Increase and Decrease**: The function increases where \( f'(x) > 0 \) and decreases where \( f'(x) < 0 \). This can be determined from the critical points found earlier. 6. **Periodicity**: The period of the sine function is \( 2\pi \). The function \( \sin(3x) \) has a period of: \[ \text{Period: } \frac{2\pi}{3} \] ### Summary - **Domain**: \( (-\infty, \infty) \) - **Range**: \( [-3, 1] \) - **Critical Points**: \( x = \frac{5\pi}{12} + \frac{k\pi}{3} \) - **Period**: \( \frac{2\pi}{3} \) If you need further analysis or specific calculations, please let me know!