Let’s analyze each of the given expressions to determine whether they represent a function for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
### 1. [tex]\( 7x = y^3 \)[/tex]
For a given value of [tex]\( x \)[/tex]:
- We can isolate [tex]\( y \)[/tex] by taking the cube root of both sides of the equation: [tex]\( y = \sqrt[3]{7x} \)[/tex].
- The cube root function produces a unique output for every input value of [tex]\( x \)[/tex].
Conclusion: For every [tex]\( x \)[/tex], there is only one corresponding value of [tex]\( y \)[/tex]. Hence, [tex]\( 7x = y^3 \)[/tex] is a function.
### 2. [tex]\( y = 7|x| - 3 \)[/tex]
For a given value of [tex]\( x \)[/tex]:
- The absolute value [tex]\( |x| \)[/tex] is a well-defined function for all real numbers and produces a unique output.
- Multiplying by 7 and then subtracting 3 will also yield a unique output value for [tex]\( y \)[/tex].
Conclusion: For every [tex]\( x \)[/tex], there is only one corresponding value of [tex]\( y \)[/tex]. Hence, [tex]\( y = 7|x| - 3 \)[/tex] is a function.
### 3. [tex]\( 36 = |y| + x^2 \)[/tex]
For a given value of [tex]\( x \)[/tex]:
- Isolate [tex]\( |y| \)[/tex]: [tex]\( |y| = 36 - x^2 \)[/tex].
- The equation [tex]\( |y| = 36 - x^2 \)[/tex] implies [tex]\( y \)[/tex] could take on two values, [tex]\( y = \pm \sqrt{36 - x^2} \)[/tex], wherever [tex]\( 36 - x^2 \geq 0 \)[/tex].
Conclusion: For certain values of [tex]\( x \)[/tex], there can be two corresponding values of [tex]\( y \)[/tex]. Hence, [tex]\( 36 = |y| + x^2 \)[/tex] is not a function.
### 4. [tex]\( 16 + y^2 = x^2 \)[/tex]
For a given value of [tex]\( x \)[/tex]:
- Isolate [tex]\( y^2 \)[/tex]: [tex]\( y^2 = x^2 - 16 \)[/tex].
- Taking the square root, we get [tex]\( y = \pm\sqrt{x^2 - 16} \)[/tex], wherever [tex]\( x^2 - 16 \geq 0 \)[/tex].
Conclusion: For certain values of [tex]\( x \)[/tex], there can be two corresponding values of [tex]\( y \)[/tex]. Hence, [tex]\( 16 + y^2 = x^2 \)[/tex] is not a function.
### Final Results:
- [tex]\( 7x = y^3 \)[/tex]: Function (1)
- [tex]\( y = 7|x| - 3 \)[/tex]: Function (1)
- [tex]\( 36 = |y| + x^2 \)[/tex]: Not a function (2)
- [tex]\( 16 + y^2 = x^2 \)[/tex]: Not a function (2)
So, the completed table will be:
[tex]\[
\begin{tabular}{|c|ll|}
\hline $7 x=y^3$ & \textbf{Function} & \\
\hline$y=7|x|-3$ & \textbf{Function} & \\
\hline $36=|y|+x^2$ & & \textbf{Not a function} \\
\hline $16+y^2=x^2$ & & \textbf{Not a function} \\
\hline
\end{tabular}
\][/tex]
