Question

Find whether following functions are one-one, onto or not:
i) f:R→R given by f(x) = x³ + 5, for all x ∈ R
ii) f:Z→Z given by f(x) = x² + 4, for all x ∈ Z

Answer 1

(i) one - one onto

explanation :- let's take two point $x_1$ and $x_2$ from domain of given function f(x) = x³ + 5 such that, $f(x_1)=f(x_2)$
or, $x_1^3+5=x_2^3+5$
or, $x_1^3=x_2^3\implies x_1=x_2$
hence, f is one - one.

range of function belongs to all real numbers because it is three degree polynomial function.

[ odd function graph is symmetrical about origin, 3 degree polynomial function is an odd function. so it's graph is symmetrical about origin and there is no any points where function is undefined. so, range of f(x) = x³ + 5 belongs to R ]

here, co -domain = Range
so, f is onto function.

(ii) neither one one nor onto
let's take two point $x_1$ and $x_2$ from domain of given function f(x) = x² + 4 such that, $f(x_1)=f(x_2)$
or, $x_1^2+4=x_2^2+4$
or,$x_1^2=x_2^2\implies x_1\neq x_2$
hence, f is not one - one function

range of f(x) $\in\mathbb{Z^+},\textbf{where,}\mathbb{Z^+}\in\{4,5,...\}$
here co -domain ≠ range
so, f is not onto