Find $$ g(0), g(-1), g(2) $$, and $$ g\left(\frac{3}{4}\right) $$ for $$ g(x)=\frac{x}{\sqrt{1-x^{2}}} $$. Find $$ g(0) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $$ g(0)=\square $$ $$ \quad $$ (Simplify your answer. Type an exact answer, using radicals as needed.) B. $$ g(0) $$ does not exist. C. $$ g(0) $$ is not a real number. Find $$ g(-1) $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $$ g(-1)-\square $$ (Simplify your answer. Type an exact annswer, using radicals as needed.) B. $$ g(-1) $$ does not exist. C. $$ g(-1) $$ is not a real number. Find $$ g(2) $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $$ g(2)=\square $$ (Simplify your answer. Type an exact answer, using radicals as needed.) B. $$ g(2) $$ does not exist. C. $$ g(2) $$ is not a real number.

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UpStudy AI · Accepted Answer

Let's evaluate the function $$ g(x) = \frac{x}{\sqrt{1 - x^2}} $$ at the specified points: $$ x = 0 $$, $$ x = -1 $$, $$ x = 2 $$, and $$ x = \frac{3}{4} $$. We'll examine each one step by step, selecting the correct choice from the provided options.

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### 1. Find $$ g(0) $$

**Evaluation:**
$$
g(0) = \frac{0}{\sqrt{1 - 0^2}} = \frac{0}{\sqrt{1}} = \frac{0}{1} = 0
$$

**Choice:**
**A. $$ g(0) = 0 $$**
  
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### 2. Find $$ g(-1) $$

**Evaluation:**
$$
g(-1) = \frac{-1}{\sqrt{1 - (-1)^2}} = \frac{-1}{\sqrt{1 - 1}} = \frac{-1}{\sqrt{0}} = \frac{-1}{0}
$$
Division by zero is undefined in real numbers.

**Choice:**
**B. $$ g(-1) $$ does not exist.**

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### 3. Find $$ g(2) $$

**Evaluation:**
$$
g(2) = \frac{2}{\sqrt{1 - 2^2}} = \frac{2}{\sqrt{1 - 4}} = \frac{2}{\sqrt{-3}}
$$
The square root of a negative number is not a real number.

**Choice:**
**C. $$ g(2) $$ is not a real number.**

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### 4. Find $$ g\left(\frac{3}{4}\right) $$

**Evaluation:**
$$
g\left(\frac{3}{4}\right) = \frac{\frac{3}{4}}{\sqrt{1 - \left(\frac{3}{4}\right)^2}} = \frac{\frac{3}{4}}{\sqrt{1 - \frac{9}{16}}} = \frac{\frac{3}{4}}{\sqrt{\frac{7}{16}}} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} = \frac{3\sqrt{7}}{7}
$$
*(Rationalized the denominator)*

**Choice:**
**A. $$ g\left(\frac{3}{4}\right) = \frac{3\sqrt{7}}{7} $$**

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**Summary of Results:**
- **$$ g(0) = 0 $$**
- **$$ g(-1) $$ does not exist.**
- **$$ g(2) $$ is not a real number.**
- **$$ g\left(\frac{3}{4}\right) = \frac{3\sqrt{7}}{7} $$**
- $$ g(0) = 0 $$ - $$ g(-1) $$ does not exist. - $$ g(2) $$ is not a real number. - $$ g\left(\frac{3}{4}\right) = \frac{3\sqrt{7}}{7} $$

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