To determine which of the given equations represent functions, recall the following:

A relation is a function if each input (x-value) corresponds to exactly one output (y-value). This can be verified using the vertical line test: if a vertical line intersects the graph of the equation more than once, the equation is not a function.

Now, let’s analyze each equation:

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1. $y = 4x + 13$:
   • This is a linear equation.
   • For every $x$, there is exactly one $y$-value.
   • This is a function.

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2. $x = 5$:
   • This is a vertical line at $x = 5$.
   • A vertical line fails the vertical line test because all points on the line have the same $x$-value but multiple $y$-values.
   • This is not a function.

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3. $x^2 + y^2 = 16$:
   • This is the equation of a circle with radius 4 centered at the origin.
   • A circle fails the vertical line test since some vertical lines intersect it at two points.
   • This is not a function.

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4. $y^2 = \frac{1}{3}x - 6$:
   • Solving for $y$, we get $y = \pm \sqrt{\frac{1}{3}x - 6}$.
   • For certain $x$-values, there are two $y$-values (positive and negative square roots).
   • This is not a function.

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5. $y = 3x^2 - x - 1$:
   • This is a quadratic equation.
   • For every $x$, there is exactly one $y$-value.
   • This is a function.

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Final Answer:

The equations that represent functions are:

1. $y = 4x + 13$
2. $y = 3x^2 - x - 1$

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Would you like a more detailed explanation of any part?

Here are 5 related questions to explore:

1. How does the vertical line test determine if a relation is a function?
2. Can a circle ever be represented as a function?
3. What are the characteristics of linear and quadratic functions?
4. How can parametric equations help describe non-functions?
5. Are there any exceptions where equations with square roots represent functions?

Tip: Always look for the presence of multiple $y$-values for a single $x$-value to identify non-functions.