Question

Find which of the functions are one-one onto, many-one, onto, one-one into, many-one into. Justify your answer:
i) f:R→R given as f(x) = 3x + 7, for all x ∈ R
ii) f:R→R given as f(x) = x² for all x ∈ R
iii) f = {(1, 3), (2, 6), (3, 9), (4, 12)} defined from A to B where A = {1, 2, 3, 4}, B = {3, 6, 9, 12, 15}

Answer 1

i) one one onto

explanation :- let's take two point $x_1$ and $x_2$ from domain of given function f(x) = 3x + 7 such that, $f(x_1)=f(x_2)$
or, $3x_1+7=3x_2+7\implies x_1=x_2$
hence, f is one - one
range of f(x) = 3x + 7 belongs to all real numbers .
here co-domain = range $\in$ R
so, f is onto function.
hence, f is one - one onto.

(ii) many - one into

explanation :- let's take two point $x_1$ and $x_2$ from domain of given function f(x) = x² such that, $f(x_1)=f(x_2)$
or, $x_1^2=x_2^2\implies x_1\neq x_2$
hence, f is not one - one so, f is many- one

range of function belongs to all positive real numbers. e.g., range $\in$ [0, ∞)

here co-domain ≠ range
so, f is not onto so, f is into.
hence, f is many- one into function.

(iii) one - one into

explanation : f is one one function because value of f for all elements of A has unique value of B.
range = {3, 6, 9, 12}
co -domain = {3, 6, 9, 12, 15}
here co-domain ≠ range
so, f is not onto so, f is into.
hence, f is one-one into