If `f(x) = (sin^2x + 4sinx + 5)/(2 sin ^2x + 8 sin x + 8)`, then range of f(x) is

Options

A. `(1/2,oo)`
B. `(5/9,1)`
C. `[ 5/9, 1]`
D. `[5/9,oo)`

Answer

To find the range of the function \( f(x) = \frac{\sin^2 x + 4 \sin x + 5}{2 \sin^2 x + 8 \sin x + 8} \), we will follow these steps:

### Step 1: Rewrite the function
We can rewrite the function by factoring out constants from the numerator and denominator:
\[ f(x) = \frac{\sin^2 x + 4 \sin x + 5}{2 \sin^2 x + 8 \sin x + 8} \]
We can factor out \(2\) from the denominator:
\[ f(x) = \frac{\sin^2 x + 4 \sin x + 5}{2(\sin^2 x + 4 \sin x + 4)} \]

### Step 2: Simplify the expression
Next, we can simplify the expression:
\[ f(x) = \frac{1}{2} \cdot \frac{\sin^2 x + 4 \sin x + 5}{\sin^2 x + 4 \sin x + 4} \]

### Step 3: Set \( y = \sin x \)
Let \( y = \sin x \). Since \( \sin x \) varies from -1 to 1, we will analyze the function in terms of \( y \):
\[ f(y) = \frac{1}{2} \cdot \frac{y^2 + 4y + 5}{y^2 + 4y + 4} \]

### Step 4: Analyze the function
Now we need to analyze the function:
\[ f(y) = \frac{1}{2} \cdot \frac{(y^2 + 4y + 4) + 1}{y^2 + 4y + 4} = \frac{1}{2} \left(1 + \frac{1}{y^2 + 4y + 4}\right) \]

### Step 5: Find the minimum and maximum values of the denominator
The denominator \( y^2 + 4y + 4 \) can be rewritten as:
\[ (y + 2)^2 \]
This expression is always non-negative and reaches its minimum value of \( 0 \) when \( y = -2 \). However, since \( y = \sin x \), the valid range for \( y \) is from -1 to 1.

Calculating the minimum and maximum values of \( (y + 2)^2 \) in the interval \( y \in [-1, 1] \):
- At \( y = -1 \): \( (-1 + 2)^2 = 1 \)
- At \( y = 1 \): \( (1 + 2)^2 = 9 \)

Thus, \( (y + 2)^2 \) varies from \( 1 \) to \( 9 \).

### Step 6: Determine the range of \( f(y) \)
Now we can find the range of \( f(y) \):
- When \( (y + 2)^2 = 1 \) (minimum value):
\[ f(y) = \frac{1}{2} \left(1 + 1\right) = 1 \]
- When \( (y + 2)^2 = 9 \) (maximum value):
\[ f(y) = \frac{1}{2} \left(1 + \frac{1}{9}\right) = \frac{1}{2} \left(\frac{10}{9}\right) = \frac{5}{9} \]

### Step 7: Conclusion
Thus, the range of \( f(x) \) is:
\[ \left[\frac{5}{9}, 1\right] \]