Question

Let x = 1, 5, 9 and y = 3, 4, 7.

a. Define f: x → y by specifying that f(1) = 4, f(5) = 7, f(9) = 4. Is f one-to-one? Is f onto? Explain your answers.
b. Define g: x → y by specifying that g(1) = 7, g(5) = 3, g(9) = 4.

Answer 1

Function f defined on the set x to y is not one-to-one because two different x values map to the same y value and is not onto because not every y value has a corresponding x value. Function g is one-to-one as every x maps to a unique y, but is also not onto since not every y value is the image of an x value under g.

Step-by-step explanation:

The student has provided two functions, f and g, mapping a set x = 1, 5, 9 to a set y = 3, 4, 7. For part a, the function f is given by f(1) = 4, f(5) = 7, f(9) = 4. This function is not one-to-one because two values of x map to the same value of y (both f(1) and f(9) map to 4). Function f is also not onto because not every element in set y is mapped to by some element in set x (the element 3 in set y is not the image of any element in x under function f).

For part b, the student defines a new function g: x → y by g(1) = 7, g(5) = 3, g(9) = 4. Each element in set x maps to a unique element in set y, making function g one-to-one. However, similar to function f, function g is not onto since the value 7 in set y is not the image of any element in x under function g.