Determine if the equation specifies a function with independent variable $\mathrm{x}$. If it does, find the domain. If it does not, find a value of $\mathrm{x}$ to which there corresponds more thar \[ x^{2}+y^{2}=144 \] Does the equation specify a function with independent value $x$ ? Yes No Based on your answer above, provide either the domain of the function or a value of $x$ to which there corresponds more than one value of $y$ A. A counterexample is $x=14$ B. The domain of the function above is all real numbers. C. A counterexample is $x=0$ D. A counterexample is $x=12$.

Step 1 ：The given equation is of a circle with radius 12. In a function, each input (or 'x' value) should correspond to exactly one output (or 'y' value). However, in the equation of a circle, each 'x' value (except the endpoints) corresponds to two 'y' values - one on the upper half of the circle and one on the lower half. Therefore, this equation does not specify a function with independent variable x.

Step 2 ：To find a counterexample, we can choose any 'x' value within the range (-12, 12) (exclusive), as these 'x' values will correspond to two 'y' values on the circle. For instance, when x=0, the corresponding 'y' values are 12 and -12. Therefore, x=0 is a counterexample.

Step 3 ：Let's confirm this by solving the equation for 'y' when x=0. x = 0, y_values = [12.0, -12.0]. As expected, when x=0, there are two corresponding 'y' values: 12 and -12. This confirms that the given equation does not specify a function with independent variable x, and x=0 is a counterexample.

Step 4 ：Final Answer: The equation does not specify a function with independent variable $x$. A counterexample is $x=0$. So, the answers are "No" and "C. A counterexample is $x=0$".