The range of the function `f(x)=x^2+1/(x^2+1)` is

Options

A. `[1, oo)`
B. `[2, oo)`
C. `[(3)/(2), oo)`
D. none of these

Answer

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

### Step 1: Simplify the function
The function can be rewritten as:
\[ f(x) = 1 \]
This is because the numerator and denominator are identical, hence the function simplifies to 1 for all \( x \) except where the denominator is zero. However, since \( x^2 + 1 \) is never zero for real \( x \), this simplification holds for all real numbers.

### Step 2: Determine the range
Since \( f(x) = 1 \) for all values of \( x \), the output of the function is always 1. Therefore, the range of the function is simply:
\[ \text{Range} = \{1\} \]

### Step 3: Conclusion
Thus, the range of the function \( f(x) = \frac{x^2 + 1}{x^2 + 1} \) is:
\[ \text{Range} = \{1\} \]

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