QuestionAugust 11, 2025

Find the range of the function. f(x)=(x^2)/(1-x^2) What is the range of f(x) square (Type your answer in interval notation.)

Find the range of the function. f(x)=(x^2)/(1-x^2) What is the range of f(x) square (Type your answer in interval notation.)
Find the range of the function. f(x)=(x^2)/(1-x^2) What is the range of f(x) square (Type your answer in interval notation.)

Solution
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Answer

(-\infty, 0) \cup (0, \infty) Explanation 1. Identify domain restrictions The denominator 1-x^2 must not be zero, so x \neq \pm 1. 2. Analyze behavior at critical points As x \to \pm 1, f(x) \to \pm \infty. 3. Determine limits as x \to \pm \infty As x \to \pm \infty, f(x) \to 0. 4. Evaluate function behavior in intervals For x \in (-\infty, -1) and (1, \infty), f(x) is positive and increases from 0 to \infty. For x \in (-1, 1), f(x) is negative and decreases from -\infty to 0. 5. Combine results for range The range of f(x) includes all real numbers except 0.

Explanation

1. Identify domain restrictions<br /> The denominator $1-x^2$ must not be zero, so $x \neq \pm 1$.<br /><br />2. Analyze behavior at critical points<br /> As $x \to \pm 1$, $f(x) \to \pm \infty$. <br /><br />3. Determine limits as $x \to \pm \infty$<br /> As $x \to \pm \infty$, $f(x) \to 0$.<br /><br />4. Evaluate function behavior in intervals<br /> For $x \in (-\infty, -1)$ and $(1, \infty)$, $f(x)$ is positive and increases from $0$ to $\infty$.<br /> For $x \in (-1, 1)$, $f(x)$ is negative and decreases from $-\infty$ to $0$.<br /><br />5. Combine results for range<br /> The range of $f(x)$ includes all real numbers except $0$.
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