Example:
Which of the following functions are one-one, onto, or bijections ? The domain
for all functions is R.
- f(x) = 2x + 5
- g(x) = arctan(x)
- g(x) = sin(x)
- h(x) = 2x3 + 5x2 - 7x + 6
1. Linear Function
This function is linear. The equation y = 2x + 5 has a unique solution
for every x, so that the function is one-one and onto, i.e. a
bijection. In fact, all linear functions are bijections.
f(x) = 2x + 5 is a bijection.
2. Inverse Functions
This function is the inverse function of the tangent function. As such, it
must be one-to-one. It is onto if the range is
[- 
/ 2, / 2],
but not if the range is R. In fact, all inverse functions are one-to-one.
g(x) = arctan(x) is injective,
not surjective on R.
3. Periodic Function
Since this function is periodic, it can not be one-to-one. It is onto, if the
range is the interval [-1, 1], but not onto if the range is R.
In fact, every periodic function is not one-to-one.
g(x) = sin(x) is neither
one-to-one nor onto R.
4. Odd Degree Polynomial
This is an odd-degree polynomial. Hence, the limit as x approaches plus
or minus infinity must be plus or minus infinity, respectively. That means
that the function is onto. Since most third degree equations have three
zero, this function is probably not one-to-one. A look at the graph confirms
this. In fact, every odd-degree polynomial is onto while no even degree
polynomial is onto.
h(x) = 2x3 + 5x2 - 7x + 6
is onto, not one-to-one
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