Question

please explainnnnnnnnnnn

Answer 1

what's a bijective function?

Answer 2

First, we must prove g is a function from B to A. This means that we have to prove g is a relation from B to A, and that for every y in B, there exists a unique x in A such that (y, x)∈g. For the first part, note that if (y, x)∈g, then (x, y)∈f⊆A×B, so (y, x)∈B×A. This means g⊆B×A, so g is a relation from B to A. Next, let y∈g be arbitrary. Since f is surjective, there exists x such that f(x) = y -- i.e. (x, y)∈f, which means (y, x)∈g. Further, if z is any other element such that (y, z)∈g, then by the definition of g, (z, y)∈f -- i.e. f(z) = y = f(x), so z=x. Thus ∀y∈B, ∃!x∈A s.t. (y, x)∈g, so g:B → A is a function.

Next, we must show that g = f⁻¹. Let x∈A be arbitrary. By the definition of function notation, (x, f(x))∈f, which by the definition of g means (f(x), x)∈g, which is to say g(f(x)) = x. Thus we have ∀x∈A, g(f(x))=x, so g∘f is the identity function on A. Similarly, let y∈B be arbitrary. Then (y, g(y))∈g, which by the definition of g implies that (g(y), y)∈f, so f(g(y)) = y. Thus ∀y∈B, f(g(y)) = y, so f∘g is the identity function on B. So g is indeed an inverse of f, and we are done with the first direction.

To prove that invertible functions are bijective, suppose f:A → B has an inverse. Let x and y be any two elements of A, and suppose that f(x) = f(y). Then x = f⁻¹(f(x)) = f⁻¹(f(y)) = y. Therefore f is injective. To show that it is surjective, let x∈B be arbitrary. Then since f⁻¹ is defined on all of B, we can let y=f⁻¹(x), so f(y) = f(f⁻¹(x)) = x. Since we can find such y for any x∈B, it follows that if is also surjective, thus bijective.

Answer 3

why u have give me so long explanation

Answer 4

all you needed to do was look on the internet.
i just c&p.

Answer 5

okay

Answer 6

All you need to know is that if a function that is not one-to-one, if you find it's inverse, it will NOT be a function, as there may be more than one y value for each x value.

A function needs to have at most one y value for each x-value.

Eg. y=x^2 and it's inverse: http://moodle.tbaisd.org/file.php/833/Quadratic/InverseOF1.png

Notice that the inverse is not a function - there are two y values for every x value (except x=0).

Answer 7

If a function fails the horizontal line test, it has no inverse function. The horizontal line test is similar to the vertical line test...

Notice that if you draw a horizontal line through y=x^2, the line will intersect the graph twice - and if you drew a vertical line through x=y^2, the line will intersect the graph twice. If it fails the vertical line test, it is not a function.

Answer 8

@agent0smith
thanks brother

Answer 9

@rosho bijective is just a fancy word for one-to-one.

Answer 10

figured that out through the internet.