In which of the following is y a function of x?I. y2=9−x2II. ∣y∣=xIII. y=x23​ I. y 2 =9−x 2 II. ∣y∣=xIII. y= 3 x 2 ​ ​

To determine whether $y$ is a function of $x$ in each case, we need to assess if each equation defines $y$ explicitly in terms of $x$.

1. Analyze Each Equation

1. Equation I: $y^2 = 9 - x^2$

   • This can be rearranged to find $y$: $y = \pm \sqrt{9 - x^2}$
   • Here, for each $x$ within the domain, there are two possible values for $y$ (one positive and one negative). Thus, $y$ is not a function of $x$.

2. Equation II: $|y| = x$

   • This results in two cases: $y = x \quad \text{or} \quad y = -x$
   • Similar to the first equation, each positive value of $x$ corresponds to two values of $y$. Hence, $y$ is not a function of $x$.

3. Equation III: $y = 3x^2$

   • This equation directly provides $y$ as a single output for each input $x$. Hence, $y$ is undoubtedly a function of $x$.

Final Answer

• Only Equation III $y = 3x^2$ is a function of $x$.