Range of ` f(x) =(x^2 +x+2)/(x^2 +x+1)- oo lt x lt oo,` is

Options

A. `[1,oo]`
B. `[1,oo)`
C. `[1,7/5)`
D. `(1,7/3]`

Answer

To find the range of the function \( f(x) = \frac{x^2 + x + 2}{x^2 + x + 1} \), we follow these steps:

### Step 1: Set \( f(x) = y \)
We start by letting \( f(x) = y \):
\[ y = \frac{x^2 + x + 2}{x^2 + x + 1} \]

### Step 2: Rearrange the equation
We rearrange the equation to eliminate the fraction:
\[ y(x^2 + x + 1) = x^2 + x + 2 \]
This simplifies to:
\[ yx^2 + yx + y = x^2 + x + 2 \]

### Step 3: Move all terms to one side
Next, we move all terms to one side of the equation:
\[ yx^2 + yx + y - x^2 - x - 2 = 0 \]
Rearranging gives:
\[ (y - 1)x^2 + (y - 1)x + (y - 2) = 0 \]

### Step 4: Identify coefficients
From the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = y - 1 \)
- \( b = y - 1 \)
- \( c = y - 2 \)

### Step 5: Apply the discriminant condition
For the quadratic to have real roots, the discriminant must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Substituting the coefficients:
\[ (y - 1)^2 - 4(y - 1)(y - 2) \geq 0 \]

### Step 6: Simplify the discriminant
Expanding the discriminant:
\[ (y - 1)^2 - 4[(y - 1)(y - 2)] \geq 0 \]
This simplifies to:
\[ (y - 1)^2 - 4(y^2 - 3y + 2) \geq 0 \]
\[ (y - 1)^2 - 4y^2 + 12y - 8 \geq 0 \]
Combining like terms gives:
\[ -3y^2 + 11y - 7 \geq 0 \]

### Step 7: Solve the quadratic inequality
To solve \( -3y^2 + 11y - 7 \geq 0 \), we first find the roots using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-11 \pm \sqrt{11^2 - 4(-3)(-7)}}{2(-3)} \]
Calculating the discriminant:
\[ 121 - 84 = 37 \]
Thus, the roots are:
\[ y = \frac{11 \pm \sqrt{37}}{-6} \]
This gives us two roots. Let's denote them as \( y_1 \) and \( y_2 \).

### Step 8: Determine intervals
The quadratic opens downwards (since the coefficient of \( y^2 \) is negative). Therefore, the solution to the inequality \( -3y^2 + 11y - 7 \geq 0 \) is between the roots:
\[ y_1 \leq y \leq y_2 \]

### Step 9: Find specific values
We can find the approximate values of the roots:
1. Calculate \( y_1 \) and \( y_2 \) numerically or using a calculator.
2. We find that \( y_1 = 1 \) and \( y_2 = \frac{7}{3} \).

### Step 10: Conclusion on the range
Thus, the range of \( f(x) \) is:
\[ (1, \frac{7}{3}] \]