Concept: Functions - Finding the Range of Rational Functions.
Step 1: Set the function equal to . Let . Therefore, . To find the range, we need to determine all possible real values that can take.
Step 2: Express in terms of . Cross-multiply to remove the fraction: .
Rearrange the equation to group all terms on one side: .
Factor out : .
Divide by to isolate : .
Step 3: Apply the condition for real . For to be a real number ( ), its square must be non-negative. This means .
Substituting our expression from Step 2, we get the inequality: .
Step 4: Solve the rational inequality. For the fraction to be greater than or equal to 0, the numerator and denominator must have the same sign. Also, the denominator cannot be zero, meaning .
Let's rewrite the inequality standard form by multiplying the numerator and denominator by -1: .
The critical points are (from the numerator) and (from the denominator).
Testing the intervals around these points using the Wavy Curve Method reveals that the inequality is satisfied when is between 0 and 1.
Step 5: Determine the exact interval bounds. Since the inequality is , we include the numerator's critical point ( ). Thus, can equal 0.
However, cannot equal 1 because it makes the original denominator zero (and undefined).
Therefore, , which is written in interval notation as . $ $