Math problem: Let f: X ?

Question

Y be a function.True or False?A sufficient condition for f to be one-to-one is that for all elements y in Ym there is at most one x in X with f(x)= y.

Answer 1

I don't know how to answer your question as a formal proof.

 

But maybe this will give you some insight.

 

A one-to-one function has an inverse that is also a function.

 

Visualizing the inverse is easy because it's the function's reflection in y=x. So a Vertical Line Test on the inverse is the same as a Horizontal Line Test on the function.

 

Any linear function is one-to-one. Horizontal Line Test gives one point.

 

A quadratic function (parabola) is not one-to-one. Horizontal Line Test may give two points.

 

Exponential and Logarithmic functions are one-to-one. Horizontal Line Test gives one point. BTW these two functions are inverses of each other.

 

All trig functions are not one-to-one. Horizontal Line Test may give infinite number of points. The arcsin, et al, functions are carefully restricted to make them one-to-one; but then the user must sometimes extend the answer to multiple points by understanding the Unit Circle.

 

Good luck.