Answer :
To determine whether [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex], we need to check if for each value of [tex]\( x \)[/tex] there is exactly one corresponding value of [tex]\( y \)[/tex]. Let's analyze each equation step by step.
1. Equation: [tex]\(3x - y = 9\)[/tex]
Analysis:
- Isolate [tex]\( y \)[/tex]:
[tex]\( y = 3x - 9 \)[/tex]
- For any value of [tex]\( x \)[/tex], there is exactly one corresponding value of [tex]\( y \)[/tex].
Conclusion: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
2. Equation: [tex]\(y = 6x^2 - 2\)[/tex]
Analysis:
- This equation is already solved for [tex]\( y \)[/tex].
- For any value of [tex]\( x \)[/tex], there is exactly one corresponding value of [tex]\( y \)[/tex].
Conclusion: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
3. Equation: [tex]\(y^2 = 4x\)[/tex]
Analysis:
- Solve for [tex]\( y \)[/tex]:
[tex]\( y = \pm \sqrt{4x} \)[/tex]
- For a given [tex]\( x \)[/tex], there are two possible values for [tex]\( y \)[/tex] (one positive and one negative).
Conclusion: [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
4. Equation: [tex]\(x = 8y^2 + 4\)[/tex]
Analysis:
- Isolate [tex]\( y \)[/tex]:
[tex]\( 8y^2 = x - 4 \)[/tex]
[tex]\( y^2 = \frac{x - 4}{8} \)[/tex]
[tex]\( y = \pm \sqrt{\frac{x - 4}{8}} \)[/tex]
- For a given [tex]\( x \)[/tex], there are two possible values for [tex]\( y \)[/tex] (one positive and one negative).
Conclusion: [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
Based on this analysis, the results for the equations are as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline 3x - y = 9 & \text{Function} & \\ \hline y = 6x^2 - 2 & \text{Function} & \\ \hline y^2 = 4x & & \text{Not a function} \\ \hline x = 8y^2 + 4 & & \text{Not a function} \\ \hline \end{array} \][/tex]
1. Equation: [tex]\(3x - y = 9\)[/tex]
Analysis:
- Isolate [tex]\( y \)[/tex]:
[tex]\( y = 3x - 9 \)[/tex]
- For any value of [tex]\( x \)[/tex], there is exactly one corresponding value of [tex]\( y \)[/tex].
Conclusion: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
2. Equation: [tex]\(y = 6x^2 - 2\)[/tex]
Analysis:
- This equation is already solved for [tex]\( y \)[/tex].
- For any value of [tex]\( x \)[/tex], there is exactly one corresponding value of [tex]\( y \)[/tex].
Conclusion: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
3. Equation: [tex]\(y^2 = 4x\)[/tex]
Analysis:
- Solve for [tex]\( y \)[/tex]:
[tex]\( y = \pm \sqrt{4x} \)[/tex]
- For a given [tex]\( x \)[/tex], there are two possible values for [tex]\( y \)[/tex] (one positive and one negative).
Conclusion: [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
4. Equation: [tex]\(x = 8y^2 + 4\)[/tex]
Analysis:
- Isolate [tex]\( y \)[/tex]:
[tex]\( 8y^2 = x - 4 \)[/tex]
[tex]\( y^2 = \frac{x - 4}{8} \)[/tex]
[tex]\( y = \pm \sqrt{\frac{x - 4}{8}} \)[/tex]
- For a given [tex]\( x \)[/tex], there are two possible values for [tex]\( y \)[/tex] (one positive and one negative).
Conclusion: [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
Based on this analysis, the results for the equations are as follows:
[tex]\[ \begin{array}{|c|c|c|} \hline 3x - y = 9 & \text{Function} & \\ \hline y = 6x^2 - 2 & \text{Function} & \\ \hline y^2 = 4x & & \text{Not a function} \\ \hline x = 8y^2 + 4 & & \text{Not a function} \\ \hline \end{array} \][/tex]
