Archangel Macsika
Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f: R—>R defined by f(x) = x2 (e) f: R—>R defined by f(x) = x3 (f) f: R—>R defined by f(x) = x – x2 (g) f: R—>R defined by f(x) = Floor(x) (h) f: R—>R defined by f(x) = Ceiling(x) (i) f: R—>R defined by f(x) = – 3x+4 (j) f: R—>R defined by f(x)= – 3x2 +7
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Solution.
We will use the Horizontal line test to determine if the function is one-to-one. If no horizontal line intersects the graph of the function f in more than one point, then the function is one-to-one.
The function is onto if codomain=range.
a)
Answer. The function is one-to-one, onto and one-to-one correspondence.
b)
Answer. The function is not one-to-one, not onto and not one-to-one correspondence.
c)
Answer. The function is one-to-one, onto and one-to-one correspondence.
d)
Answer. The function is not one-to-one, not onto and one-to-one correspondence.
e)
Answer. The function is one-to-one, onto and one-to-one correspondence.
f)
Answer. The function is not one-to-one, not onto and not one-to-one correspondence.
g)
Answer. The function is not one-to-one, not onto and not one-to-one correspondence.
h)
Answer. The function is not one-to-one, not onto and not one-to-one correspondence.
i)
Answer. The function is one-to-one, onto and one-to-one correspondence.
j)
Answer. The function is not one-to-one, not onto and not one-to-one correspondence.