Bijective function

Elementary functions

Let f(x):ℝ→ℝ be a real-valued function y=f(x) of a real-valued argument x. (This means both the input and output are numbers.)

• Graphic meaning: The function f is a bijection if every horizontal line intersects the graph of f in exactly one point.
• Algebraic meaning: The function f is a bijection if for every real number y_o we can find at least one real number x_o such that y_o=f(x_o) and if f(x_o)=f(x₁) means x_o=x₁ .

Example: The linear function of a slanted line is a bijection. That is, y=ax+b where a≠0 is a bijection.

Discussion: Every horizontal line intersects a slanted line in exactly one point (see surjection and injection for proofs). Image 1.

Example: The polynomial function of third degree: f(x)=x³ is a bijection. Image 2 and image 5 thin yellow curve. Its inverse is the cube root function f(x)= ∛x and it is also a bijection f(x):ℝ→ℝ. Image 5: thick green curve.

Example: The quadratic function f(x) = x² is not a bijection (from ℝ→ℝ). Image 3. It is not a surjection. It is not an injection. However, we can restrict both its domain and codomain to the set of non-negative numbers (0,+∞) to get an (invertible) bijection (see examples below).

• the domain
• the function machine
• the codomain

Example: Suppose our function machine is f(x)=x².

• This machine and domain=ℝ and codomain=ℝ is not a surjection and not an injection. However,
• this same machine and domain=[0,+∞) and codomain=[0,+∞) is both a surjection and an injection and thus a bijection.

Bijections and their inverses

Let f(x):A→B where A and B are subsets of ℝ.

• Suppose f is not a bijection. For any x where the derivative of f exists and is not zero, there is a neighborhood of x where we can restrict the domain and codomain of f to be a bijection.^: 281
• The graphs of inverse functions are symmetric with respect to the line y=x. (See also Inverse function.)

Example: The quadratic function defined on the restricted domain and codomain [0,+∞)

${\displaystyle f(x):[0,+\infty )\,\,\rightarrow \,\,[0,+\infty )}$  defined by  ${\displaystyle f(x)=x^{2}}$

is a bijection. Image 6: thin yellow curve.

Example: The square root function defined on the restricted domain and codomain [0,+∞)

${\displaystyle f(x):[0,+\infty )\,\,\rightarrow \,\,[0,+\infty )}$  defined by  ${\displaystyle f(x)={\sqrt {x}}}$

is the bijection defined as the inverse function of the quadratic function: x². Image 6: thick green curve.

${\displaystyle f(x):\mathbf {R} \,\,\rightarrow \,\,(0,+\infty )}$  defined by  ${\displaystyle f(x)=a^{x}\,,\,\,a>1}$

is a bijection. Image 4: thin yellow curve (a=10).

Example: The logarithmic function base a defined on the restricted domain (0,+∞) and the codomain ℝ

${\displaystyle f(x):(0,+\infty )\,\,\rightarrow \,\,\mathbf {R} }$  defined by  ${\displaystyle f(x)=\log _{a}x\,,\,\,a>1}$

is the bijection defined as the inverse function of the exponential function: a^x. Image 4: thick green curve (a=10).

Bijection: every vertical line (in the domain) and every horizontal line (in the codomain) intersects exactly one point of the graph.
1. Bijection. All slanted lines are bijections f(x):ℝ→ℝ.	2. Bijection. f(x):ℝ→ℝ. f(x)=x³.	3. Not a bijection. f(x):ℝ→ℝ. f(x)=x² is not a surjection. It is not an injection.
4. Bijections. f(x):ℝ→ (0,+∞). f(x)=10x (thin yellow) and its inverse f(x):(0,+∞)→ℝ. f(x)=log10x (thick green).	5. Bijections. f(x):ℝ→ℝ. f(x)=x³ (thin yellow) and its inverse f(x)=∛x (thick green).	6. Bijections. f(x):[0,+∞)→[0,+∞). f(x)=x² (thin yellow) and its inverse f(x)=√x (thick green).