Bijection - More Mathematical Examples and Some Non-examples

More Mathematical Examples and Some Non-examples

• For any set X, the identity function 1_X: X → X, 1_X(x) = x, is bijective.
• The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. In more generality, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.
• The function f: R → (-π/2, π/2), given by f(x) = arctan(x) is bijective since each real number x is paired with exactly one angle y in the interval (-π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain (-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto (surjective) since there is no real number which could be paired with the multiple of π/2 by this arctan function.
• The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbers, then g becomes bijective; its inverse (see below) is the natural logarithm function ln.
• The function h: R → R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to, then h becomes bijective; its inverse is the positive square root function.