Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Determine if each of the following equations represents a function:

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Equation & Function & Not a Function \\
\hline
$7x = y^3$ & & \\
\hline
$y = 7|x| - 3$ & & \\
\hline
$36 = |y| + x^2$ & & \\
\hline
$16 + y^2 = x^2$ & & \\
\hline
\end{tabular}
\][/tex]


Sagot :

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]