Injective 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 real numbers.)

• Graphic meaning: The function f is an injection if every horizontal line intersects the graph of f in at most one point.
• Algebraic meaning: The function f is an injection if f(x_o)=f(x₁) means x_o=x₁.

Example: The linear function of a slanted line is 1-1. That is, y=ax+b where a≠0 is an injection. (It is also a surjection and thus a bijection.)

Proof: Let x_o and x₁ be real numbers. Suppose the line maps these two x-values to the same y-value. This means a·x_o+b=a·x₁+b. Subtract b from both sides. We get a·x_o=a·x₁. Now divide both sides by a (remember a≠0). We get x_o=x₁. So we have proved the formal definition and the function y=ax+b where a≠0 is an injection.

Example: The polynomial function of third degree: f(x)=x³ is an injection. However, the polynomial function of third degree: f(x)=x³ –3x is not an injection.

Discussion 1: Any horizontal line intersects the graph of

f(x)=x³ exactly once. (Also, it is a surjection.)

Discussion 2. Any horizontal line between y=-2 and y=2 intersects the graph in three points so this function is not an injection. (However, it is a surjection.)

Example: The quadratic function f(x) = x² is not an injection.

Discussion: Any horizontal line y=c where c>0 intersects the graph in two points. So this function is not an injection. (Also, it is not a surjection.)

Note: One can make a non-injective function into an injective function by eliminating part of the domain. We call this restricting the domain. For example, restrict the domain of f(x)=x² to non-negative numbers (positive numbers and zero). Define

${\displaystyle f_{/[0,+\infty )}(x):[0,+\infty )\rightarrow \mathbf {R} }$  where  ${\displaystyle f_{/[0,+\infty )}(x)=x^{2}}$

This function is now an injection. (See also restriction of a function.)

Example: The exponential function f(x) = 10^x is an injection. (However, it is not a surjection.)

Discussion: Any horizontal line intersects the graph in at most one point. The horizontal lines y=c where c>0 cut it in exactly one point. The horizontal lines y=c where c≤0 do not cut the graph at any point.

Note: The fact that an exponential function is injective can be used in calculations.

${\displaystyle a^{x_{0}}=a^{x_{1}}\,\,\Rightarrow \,\,x_{0}=x_{1},\,a>0}$ 

Example:  ${\displaystyle 100=10^{x-3}\,\,\Rightarrow \,\,2=x-3\,\,\Rightarrow \,\,x=5}$ 

Injection: no horizontal line intersects more than one point of the graph
Injection. f(x):ℝ→ℝ (and surjection)	Injection. f(x):ℝ→ℝ (and surjection)	Not an injection. f(x):ℝ→ℝ (is surjection)
Not an injection. f(x):ℝ→ℝ (not surjection)	Injection. f(x):ℝ→ℝ (not surjection)	Injection. f(x):(0,+∞)→ℝ (and surjection)

Other examples

Example: The logarithmic function base 10 f(x):(0,+∞)→ℝ defined by f(x)=log(x) or y=log₁₀(x) is an injection (and a surjection). (This is the inverse function of 10^x.)

Example: The function f:ℕ→ℕ that maps every natural number n to 2n is an injection. Every even number has exactly one pre-image. Every odd number has no pre-image.