Question

I am a bit confused on this one if someone can please help. Suppose thaty f is a rule of correspondence with domain elements x,y, z, and t. The rule is ;x(arrow right)1, z(arrow right)1, t(arrow right)1. Is f a function? yes or no and why?

Answer 1

Did you exclude y->1 by accident? Can a function not have a mapping for a domain element? If so, then it's a function because each range element has a unique domain element.

Answer 2

a and d are *not* functions. b and c *are* functions. A function can't map one value to multiple values. You can test this easily by looking to see if a vertical line will intersect two points on the graph. For graph a, the y-axis hits two points on the circle, so it's not a function. For graph d, it maps one x value to 3 discrete y values, so a vertical line will hit all of those points, so it's not a function. Graph b is a function because there is exactly one y for each x. Graph c is a function, even though it doesn't extend to negative x's. It doesn't try to map one x value to two y values.

Even though these examples use a graph on a plane, the same ideas apply to your more abstract example. Think about it as if the domain was a set of x-values, and the range was a set of y-values. What would the graph look like? Does it pass the vertical line test?