give an example of a functional which is neither one-one nor onto
aryansingh12:
Here are a couple examples: This graph shows a function, because there is no vertical line that will cross this graph twice. This graph does not show a function, because any number of vertical lines will intersect this oval twice. For instance, the y-axis intersects (crosses) the line twice.
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1) [10 points] Give examples of functions f : R → R such that:
(a) f is one-to-one, but not onto.
Solution. There are many examples, for instance, f(x) = ex
. We know that it is
one-to-one and onto (0,∞), so it is one-to-one, but not onto all of R.
(b) f is onto, but not one-to-one.
Solution. There are many examples, for instance,
f(x) = (
ln(x), if x > 0,
0, if x ≤ 0.
We know that ln(x) is onto, as it is the inverse of ex
: R → (0,∞). But it’s domain
is not R. We make the domain R by “attaching” the half-line from (−∞, 0] at y = 0.
Then, its not one-to-one, as f(−1) = f(−2) = 0.
(c) f is neither one-to-one nor onto.
Solution. There are many examples, for instance, f(x) = x
2
. Not onto, since the image
of f(x) is [0,∞), and not one-to-one, since f(−1) = f(1).
(a) f is one-to-one, but not onto.
Solution. There are many examples, for instance, f(x) = ex
. We know that it is
one-to-one and onto (0,∞), so it is one-to-one, but not onto all of R.
(b) f is onto, but not one-to-one.
Solution. There are many examples, for instance,
f(x) = (
ln(x), if x > 0,
0, if x ≤ 0.
We know that ln(x) is onto, as it is the inverse of ex
: R → (0,∞). But it’s domain
is not R. We make the domain R by “attaching” the half-line from (−∞, 0] at y = 0.
Then, its not one-to-one, as f(−1) = f(−2) = 0.
(c) f is neither one-to-one nor onto.
Solution. There are many examples, for instance, f(x) = x
2
. Not onto, since the image
of f(x) is [0,∞), and not one-to-one, since f(−1) = f(1).
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