Question

Determine if the given equations are functions or not functions.

a) x = 3
a) y = -x + 11
c) x² + y² = 81
d) y² = -5x - 12

Answer 1

The given equations are evaluated using the vertical line test. Equations x = 3 and x² + y² = 81 are not functions, while y = -x + 11 is a function. y² = -5x - 12 is also not a function because it does not produce a unique output for each input.

Step-by-step explanation:

To determine if the given equations are functions or not, we need to check whether each equation passes the vertical line test. A vertical line test checks if any vertical line drawn through the graph of the equation intersects the graph more than once. If it does intersect more than once, then the equation is not a function.

1. x = 3 is a vertical line that will pass the vertical line test because it doesn't intersect any other point at x = 3. Hence, it is not a function.
2. y = -x + 11 is a linear equation which is a function as it will pass the vertical line test. Any vertical line will intersect the graph only once.
3. The equation x² + y² = 81 represents a circle with radius 9. A vertical line test would intersect the circle at two points in most cases, indicating that it is not a function.
4. Lastly, y² = -5x - 12 does not define y as a unique value for each x, implying that for some values of x, there are two possible values of y. Therefore, it is not a function.