Find \( g(0), g(-5), g(7) \), and \( g\left(\frac{3}{2}\right) \) for \( g(x)=\frac{x}{\sqrt{25-x^{2}}} \). 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. A. \( g(-5) \). Select the correct choice below and, if necessary, fill in the answor box to complete your choice. B. \( \quad( \) Simplify your answer. Type an exact answer, using radicals as needed.) B( -5\( ) \) does not exist. C. \( g(-5) \) is not a real nurnber. Find \( g(7) \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Let's evaluate the function \( g(x) = \frac{x}{\sqrt{25 - x^{2}}} \) at the points \( x = 0 \), \( x = -5 \), \( x = 7 \), and \( x = \frac{3}{2} \). We'll determine whether each value exists and, if it does, simplify the expression. --- ### 1. Find \( g(0) \) **Evaluation:** \[ g(0) = \frac{0}{\sqrt{25 - (0)^2}} = \frac{0}{\sqrt{25}} = \frac{0}{5} = 0 \] **Options:** - **A.** \( g(0) = \boxed{0} \) - B. \( g(0) \) does not exist. - C. \( g(0) \) is not a real number. **Correct Choice:** **A.** \( g(0) = \boxed{0} \) --- ### 2. Find \( g(-5) \) **Evaluation:** \[ g(-5) = \frac{-5}{\sqrt{25 - (-5)^2}} = \frac{-5}{\sqrt{25 - 25}} = \frac{-5}{\sqrt{0}} = \frac{-5}{0} \] Division by zero is undefined. **Options:** - A. \( g(-5) = \square \) \( \quad(\) Simplify your answer. Type an exact answer, using radicals as needed.) - **B.** \( g(-5) \) does not exist. - C. \( g(-5) \) is not a real number. **Correct Choice:** **B.** \( g(-5) \) does not exist. --- ### 3. Find \( g(7) \) **Evaluation:** \[ g(7) = \frac{7}{\sqrt{25 - 7^2}} = \frac{7}{\sqrt{25 - 49}} = \frac{7}{\sqrt{-24}} \] The square root of a negative number is not a real number. **Options:** - A. \( g(7) = \square \) \( \quad(\) Simplify your answer. Type an exact answer, using radicals as needed.) - B. \( g(7) \) does not exist. - **C.** \( g(7) \) is not a real number. **Correct Choice:** **C.** \( g(7) \) is not a real number. --- ### 4. Find \( g\left(\frac{3}{2}\right) \) **Evaluation:** \[ g\left(\frac{3}{2}\right) = \frac{\frac{3}{2}}{\sqrt{25 - \left(\frac{3}{2}\right)^2}} = \frac{\frac{3}{2}}{\sqrt{25 - \frac{9}{4}}} = \frac{\frac{3}{2}}{\sqrt{\frac{100}{4} - \frac{9}{4}}} = \frac{\frac{3}{2}}{\sqrt{\frac{91}{4}}} = \frac{\frac{3}{2}}{\frac{\sqrt{91}}{2}} = \frac{3}{\sqrt{91}} \] To rationalize the denominator: \[ \frac{3}{\sqrt{91}} \times \frac{\sqrt{91}}{\sqrt{91}} = \frac{3\sqrt{91}}{91} \] **Options:** - A. \( g\left(\frac{3}{2}\right) = \boxed{\dfrac{3\sqrt{91}}{91}} \) - B. \( g\left(\frac{3}{2}\right) \) does not exist. - C. \( g\left(\frac{3}{2}\right) \) is not a real number. **Correct Choice:** **A.** \( g\left(\frac{3}{2}\right) = \boxed{\dfrac{3\sqrt{91}}{91}} \) --- ### Summary of Results - \( g(0) = 0 \) - \( g(-5) \) does not exist. - \( g(7) \) is not a real number. - \( g\left(\frac{3}{2}\right) = \dfrac{3\sqrt{91}}{91} \) If you have any further questions or need more assistance with these calculations, feel free to ask!