Answer to Question #209903 in Discrete Mathematics for Sach

Question

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one corre￾spondence.

(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

Answer 1

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.