# One-to-One Function

For non-empty sets A and B f : If the function value of each different element in the domain of the function A → B is different, the function f is called a one-to-one function.
For ∀a, b ∈ A, f is one-to-one if the condition a ≠ b ⇒ f(a) ≠ f(b) or f(a) = f(b) ⇒ a = b is met.

In other words, if f(a)=f(b) equality can be achieved for any two different values in the domain (let it be a, b), the function is not one-to-one. Another way to understand this is to look at the graph of the function:
Since a parallel (red) drawn to the x-axis in the above graph can intersect the f(x) = sin(x) function (blue) at more than one point, the function is not one-to-one. However, since it is not possible to draw a line parallel to the x-axis and intersecting f(x) = log(x) at more than one point in the graph below, the function is one-to-one.

Click to draw function graphs.

Non-one-to-one functions are functions that cannot be reversed because:
Let f(x) be a non-one-to-one function. So it is possible to find values of a and b that are not equal to each other and satisfy f(a) = f(b). Let g(x) be the inverse of f(x). g(f(a)) = a and g(f(b)) = b. However, the equation g(f(a)) = g(f(b)) = a = b cannot be true because the values of a and b are different from each other. Thus, we prove by contradiction that non-one-to-one functions are not reversible for the entire domain. In the range in which they are reversible, there should not be different values of a and b, which satisfy the equation f(a) = f(b).

###### All Articles

What is an Online Graphics Plotter? Plotting in 3-D What are Simple Calculators and Where Are They Used? What is a Scientific Calculator and What Is It Used For? Calculator Designs And Types The Change of Calculator from History to the Present The First Mechanical Calculator Online Scientific Calculator Online Standard Simple Calculator What is Scientific Calculator? One-to-One Function What is function? Who is Euclid? Our Important Transactional Tool: Trigonometry - Triangle Mathematics/Geometry Riemann - His Life, Work and Integral Topic The World's Most Advanced Calculator What is Integral? Programmable Calculators What is Calculator Software? Where is it used?

id: HoVrKYDh75