What is a Bijective Function and How To Recognize It?

Properties of Bijective Functions

1. Invertibility
   Since  $f$ is both one-to-one and onto, there will exist a unique inverse $f^{-}:B\to A$ with $f^{-}\left(f\right(x\left)\right)=x$ , $f\left(f^{-}\right(y\left)\right)=y$.
2. One-to-One (Injective)
   Unique inputs result in unique outputs: $\forall x_1,x_2\in A,f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$.
3. Onto (Surjective)
   Each element of codomain is achieved: $\forall y\in B,\exists x\in A:f\left(x\right)=y$.
4. Cardinality Preservation
   A and B both have the same number of elements, even for infinite sets.
5. Composition Closure
   Composition of 2 bijections is a bijection. If $f:A\to B$ and $g:B\to C$ are bijections. Then  $g\circ f:A\to C$ will also be bijective, with ${g\circ f}^{-}=f^{-}\circ g^{-}$.
6. Permutation Behavior (Finite Sets)
   In a finite set, each bijection is a permutation. It rearranges all elements with no repeats and no omissions.
7. Graph Symmetry
   Graph of $f^{-}$ is the reflection of graph of $f$ across the line $y=x$.

1. Is the Function Injective (One-to-One)?

Let us prove that

$$f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$$

A. Direct algebraic proof

1. Let us assume that  $f\left(x_1\right)=f\left(x_2\right)$.
2. Solve the equation using algebra until you get  $x_1=x_2$.
3. Conclude that no two inputs will ever share the same output.

For Example:

$f\left(x\right)=3x+2$

Let us assume that $3x_1+2=3x_2+2$.

We will first subtract 2 and then divide it by 3 

Finally, we will get:  $x_1=x_2$.

B. Alternative viewpoints

• Monotonic functions on $ℝ$: If $f\prime \left(x\right)>0$ everywhere.  In this case, f is injective.
• Polynomials: A non-constant polynomial of odd degree over $ℝ$ might not come as injective only at the turning point. You will be required to check the derivative sign or else use factorization.

2. Is the function Surjective (Onto)?

Let us prove that:

$$\forall y\in B,\exists x\in A:f\left(x\right)=y$$

A.  Let us solve for x first

1. Choose an arbitrary $y\in B$.
2. Solve the equation $f\left(x\right)=y$ for x .
3. Verify whether your solution x is lying in the domain A .

For example: $f\left(x\right)=3x+2$.

Given y ,

solve $y=3x+2$

$x=\frac{y-2}{3}$

Since $\frac{y-2}{3}\in ℝ$, every y has a pre‐image.