Thread: still ont to ont problem

(a). Define the function f:

by f(x,y) = (x+2y, 3x+4y). Is f one-to-one?
(Justify your answer)

can i answer this question like follow
x+2y=x+2y
x+y=x+y

3x+4y=3x+4y
x+y=x+y

?i know sounds stupid but this is all i can remember from the lecture.
anyone can give me some ideas? thanks^0^

one to one problem~~~... i mean

Have you studied matrices?

What you can do is express the simultaneous equations in matrix form. As it's a simple 2x2 it's then straightforward to determine whether the determinant is zero.

If it is not zero, then the matrix is invertible, and therefore the function is one-to-one and onto.

Any help?

Originally Posted by paulzho89

(a). Define the function f:

by f(x,y) = (x+2y, 3x+4y). Is f one-to-one?
(Justify your answer)

can i answer this question like follow
x+2y=x+2y
x+y=x+y

3x+4y=3x+4y
x+y=x+y

?i know sounds stupid but this is all i can remember from the lecture.
anyone can give me some ideas? thanks^0^

Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.

From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?

Originally Posted by HallsofIvy

Do you remember the definition of "one to one function"? A function is "one to one" if and only if f(a)= f(b) only when a= b. Since here, f is applied to two numbers, that is "f(a,b)= f(x,y) if and only if (a,b)= (x,y)" which is the same as saying a= x and b= y.

From f(a,b)= (a+ 2b, 3a+ 4b)= (x+ 2y, 3x+ 4y)= f(x,y), which is the same as saying a+ 2b= x+ 2y and 3a+ 4b= 3x+ 4y, can you show that a= x and b= y?

thanks ..help me out^.^