select the correct answer from each drop - down menu. the graph of the function f(x) = 3/4 sin(x)+1 is…

Explanation:

Step1: Recall range of sine function

The range of $y = \sin(x)$ is $[- 1,1]$.

Step2: Find maximum value of $f(x)$

For $f(x)=\frac{3}{4}\sin(x)+1$, when $\sin(x) = 1$, $f(x)=\frac{3}{4}\times1 + 1=\frac{3 + 4}{4}=\frac{7}{4}=1.75$.

Step3: Find minimum value of $f(x)$

When $\sin(x)=-1$, $f(x)=\frac{3}{4}\times(-1)+1=1-\frac{3}{4}=\frac{1}{4}=0.25$.

Step4: Analyze function on $(0,\frac{\pi}{2})$

The derivative of $y = \sin(x)$ is $y'=\cos(x)$. For $f(x)=\frac{3}{4}\sin(x)+1$, its derivative $f'(x)=\frac{3}{4}\cos(x)$. On the interval $(0,\frac{\pi}{2})$, $\cos(x)>0$, so $f'(x)=\frac{3}{4}\cos(x)>0$, and the function is increasing.

Step5: Determine range of $f(x)$

Since the minimum value of $f(x)$ is $0.25$ and the maximum is $1.75$, the range is $[0.25,1.75]$. But if we consider the general - form based on the transformation of the sine - function, when $\sin(x)=-1,f(x)=\frac{3}{4}\times(-1)+1=\frac{1}{4}$ and when $\sin(x) = 1,f(x)=\frac{3}{4}\times1+1=\frac{7}{4}$. The range of $y = f(x)$ is $[\frac{1}{4},\frac{7}{4}]=[0.25,1.75]$.

Answer:

The maximum value of the function is $1.75$. The minimum value of the function is $0.25$. On the interval $(0,\frac{\pi}{2})$, the function is increasing. The range of the function is $[0.25,1.75]$.