Prove that the function f is surjective or injective or both

To prove that a function f is surjective or injective or both, you need to use the definitions of these terms and show that they are satisfied for the given function. Here are some general steps to follow:

• To prove that f is injective, you need to show that if

  f(x) = f(y)

  then

  x = y

  for any

  x

  and

  y

  in the domain of f. This means that f maps different inputs to different outputs. You can also use the horizontal line test to check if f is injective: a horizontal line should never intersect the graph of f at two or more points.
• To prove that f is surjective, you need to show that for any

  y

  in the codomain of f, there exists an

  x

  in the domain of f such that

  f(x) = y

  . This means that f maps every element of the codomain to some element of the domain. You can also use the intermediate value theorem to check if f is surjective: if f is continuous and its domain is an interval, then f takes all values between its minimum and maximum.
• To prove that f is bijective, you need to show that f is both injective and surjective. This means that f maps every element of the domain to a unique element of the codomain, and vice versa. A bijective function has an inverse that is also a function.

For example, let’s consider the function

f(x) = x^2

from the set of real numbers to the set of non-negative real numbers. Is f injective, surjective, or bijective?

• To check if f is injective, we assume that

  f(x) = f(y)

  and try to show that

  x = y

  . We have

  x^2 = y^2

  , which implies that

  x = \pm y

  . This means that f is not injective, because we can have different values of x and y that give the same output, such as

  f(2) = f(-2) = 4

  . We can also see this from the graph of f, which fails the horizontal line test.
• To check if f is surjective, we pick any

  y

  in the codomain of f and try to find an

  x

  in the domain of f such that

  f(x) = y

  . We have

  y = x^2

  , which implies that

  x = \pm \sqrt{y}

  . This means that f is surjective, because we can always find a value of x that gives any value of y, as long as y is non-negative. We can also see this from the graph of f, which takes all values from 0 to infinity.
• To check if f is bijective, we need to show that f is both injective and surjective. Since we already found that f is not injective, we can conclude that f is not bijective. We can also see this from the fact that f does not have an inverse that is a function, because the inverse relation

  y = \pm \sqrt{x}

  is not a function.

I hope this helps you understand how to prove that a function is surjective or injective or both. If you have a specific function that you want me to check, please let me know. 😊