$$ y = \frac { \sqrt { 4 - | x | } } { x } $$

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Certainly! Let's analyze the function: $$ y = \frac{ \sqrt{4 - |x|} }{x } $$ ### 1. **Domain of the Function** To determine the domain, we need to consider the restrictions imposed by both the square root and the denominator. - **Square Root Restriction:** The expression inside the square root must be non-negative: $$ 4 - |x| \geq 0 \implies |x| \leq 4 \implies -4 \leq x \leq 4 $$ - **Denominator Restriction:** The denominator cannot be zero: $$ x \neq 0 $$ **Combined Domain:** $$ x \in [-4, 0) \cup (0, 4] $$ ### 2. **Behavior and Asymptotes** - **Vertical Asymptote:** At $$ x = 0 $$, the denominator approaches zero, which means the function has a vertical asymptote at $$ x = 0 $$. - **Behavior Near the Vertical Asymptote:** - As $$ x \to 0^+ $$: $$ y = \frac{ \sqrt{4 - 0} }{0^+} = \frac{2}{0^+} = +\infty $$ - As $$ x \to 0^- $$: $$ y = \frac{ \sqrt{4 - 0} }{0^-} = \frac{2}{0^-} = -\infty $$ - **Horizontal Behavior:** - As $$ x \to 4^- $$ or $$ x \to -4^+ $$: $$ y = \frac{ \sqrt{4 - 4} }{x} = \frac{0}{x} = 0 $$ So, the function approaches 0 as $$ x $$ approaches 4 or -4. ### 3. **Symmetry of the Function** - The function can be tested for even, odd, or neither symmetry by evaluating $$ y(-x) $$: $$ y(-x) = \frac{ \sqrt{4 - |-x|} }{ -x } = \frac{ \sqrt{4 - |x|} }{ -x } = -\frac{ \sqrt{4 - |x|} }{ x } = -y(x) $$ - **Conclusion:** The function is **odd**, meaning it has rotational symmetry about the origin (180° rotation). ### 4. **Graphical Representation** Here's a qualitative sketch of the function: - **For $$ x > 0 $$:** The function is positive since both the numerator and denominator are positive. It starts from $$ y \to +\infty $$ as $$ x \to 0^+ $$ and decreases towards 0 as $$ x \to 4^- $$. - **For $$ x < 0 $$:** The function is negative since the numerator is positive and the denominator is negative. It starts from $$ y \to -\infty $$ as $$ x \to 0^- $$ and increases towards 0 as $$ x \to -4^+ $$. - **Endpoints:** At $$ x = \pm 4 $$, $$ y = 0 $$. ### 5. **Summary of Key Features** - **Domain:** $$ x \in [-4, 0) \cup (0, 4] $$ - **Vertical Asymptote:** $$ x = 0 $$ - **Horizontal Behavior:** $$ y \to 0 $$ as $$ x \to \pm 4 $$ - **Symmetry:** Odd function (rotational symmetry about the origin) - **Intercepts:** - **Y-Intercept:** None (since $$ x \neq 0 $$) - **X-Intercepts:** $$ (4, 0) $$ and $$ (-4, 0) $$ ### 6. **Graph Sketch** While I can't provide a visual graph here, you can visualize the function based on the description: - **Right Branch ($$ x > 0 $$):** Starts from positive infinity at $$ x \to 0^+ $$ and curves downward towards the point $$ (4, 0) $$. - **Left Branch ($$ x < 0 $$):** Starts from negative infinity at $$ x \to 0^- $$ and curves upward towards the point $$ (-4, 0) $$. ### 7. **Example Values** To better understand the function, let's calculate some example points: - **For $$ x = 1 $$:** $$ y = \frac{ \sqrt{4 - 1} }{1 } = \frac{ \sqrt{3} }{1 } \approx 1.732 $$ - **For $$ x = 2 $$:** $$ y = \frac{ \sqrt{4 - 2} }{2 } = \frac{ \sqrt{2} }{2 } \approx 0.707 $$ - **For $$ x = -1 $$:** $$ y = \frac{ \sqrt{4 - 1} }{ -1 } = \frac{ \sqrt{3} }{ -1 } \approx -1.732 $$ - **For $$ x = -2 $$:** $$ y = \frac{ \sqrt{4 - 2} }{ -2 } = \frac{ \sqrt{2} }{ -2 } \approx -0.707 $$ These points can help you plot the function more accurately. ### 8. **Conclusion** The function $$ y = \frac{ \sqrt{4 - |x|} }{x } $$ is defined for all real numbers $$ x $$ where $$ -4 \leq x \leq 4 $$ and $$ x \neq 0 $$. It has a vertical asymptote at $$ x = 0 $$ and approaches zero as $$ x $$ approaches $$ \pm 4 $$. The function is odd, exhibiting rotational symmetry about the origin. If you have a specific question or aspect of this function you'd like to explore further, feel free to ask! The function $$ y = \frac{ \sqrt{4 - |x|} }{x } $$ is defined for $$ x $$ between -4 and 4, excluding 0. It has a vertical asymptote at $$ x = 0 $$ and approaches 0 as $$ x $$ approaches -4 or 4. The function is odd, meaning it is symmetric about the origin.

\( y = \frac { \sqrt { 4 - | x | } } { x } \)

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