Answer to Question #320911 in Discrete Mathematics for Bryan

Question

Given g(x) = (x + 1)(x^2 − x), g: R → R where R is the set of real numbers.

a) Find the domain and range of the function g. (2 marks)

b) Determine whether the function is injective, surjective, and/or bijective. Justify

your answers. (7 marks)

Answer 1

$a:\\Dom\left( g \right) =\mathbb{R} , \sin ce\,\,g\,\,is\,\,defined\,\,for\,\,all\,\,x\\\underset{x\rightarrow +\infty}{\lim}\left( x+1 \right) \left( x^2-x \right) =\left[ +\infty \cdot +\infty \right] =+\infty \\\underset{x\rightarrow -\infty}{\lim}\left( x+1 \right) \left( x^2-x \right) =\left[ +\infty \cdot -\infty \right] =-\infty \\Since\,\,g\,\,is\,\,continuous, from\,\,these\,\,Range\left( g \right) =\mathbb{R} \\b:\\injective\,\,-\,\,false: g\left( 0 \right) =g\left( 1 \right) =0\\surjective\,\,-\,\,true, \sin ce\,\,Range\left( g \right) =\mathbb{R} \\bijective\,\,-\,\,false, \sin ce\,\,it\,\,is\,\,not\,\,injective$