For what values of k the relation R:{(k2 , 5), (3k, 6)} is not a function?

Explanation:

If you have repeated x values, then you wont have a function. For example, the points (1,5) and (1,6) mean we don't have a function since the input x = 1 leads to multiple outputs y = 5 and y = 6 simultaneously. For a function to be possible, we must have any input lead to exactly one output only.

What we do is set the x coordinates (k^2 and 3k) equal to each other and solve for k

k^2 = 3k

k^2-3k = 0

k(k-3) = 0 factoring

k = 0 or k-3 = 0 zero product property

k = 0 or k = 3

If k = 0, then (k^2,5) becomes (0,5). Also, (3k,6) becomes (0,6). The two points (0,5) and (0,6) mean the graph fails the vertical line test.

Similarly, if k = 3, then (k^2,5) becomes (9,5) and (3k,6) = (9,6). Another vertical line test failure happens here to show we don't have a function.