Review the definitions for one-to-one and onto functions. How are they related?
False
Practice makes perfect
If we review the definitions for each type of function, we can see that these are different and independent concepts. Therefore, one does not imply the other.
A function is one-to-one if each element of the domain pairs to exactly one unique element of the range.
A function is onto if each element of the range corresponds to an element of the domain.
Notice that we can have an onto function which is not one-to-one if all elements of the domain correspond to an element of the range and m
ore than one element of the domain corresponds to the same element of the range. For example, let's consider the function f(x) = x +2Sin(x).
<jsxgpre id="Solution81055_0" static=1>
var b=mlg.board([-17,15,17,-15],{"desktopSize":"medium","style":"usa"});
b.xaxis(2,0
);
b.yaxis(2,0);
var func1 = b.func("x+2*sin(x)");
</jsxgpre>
From the graph we can see that there are several x values with the same y value — they are the same height in the graph. Nevertheless, for each y value there is always a corresponding x value. Hence, the statement of the exercise is false.
