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Maths-

General

Easy

Question

The function $f colon R not stretchy rightwards arrow R$ given by $f left parenthesis x right parenthesis equals x cubed minus 1$ is

A one-one function
A one-one function
A bijection
Neither one-one nor onto

The correct answer is: A bijection

Given, $f left parenthesis x right parenthesis equals x plus square root of x squared end root$
Since, this function is not defined
Let $ϕ left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis$
$equals open curly brackets table attributes columnspacing 1em end attributes row cell x comma x element of calligraphic Q end cell row cell negative x comma x not an element of calligraphic Q end cell end table close$
For one-one
Take any straight line parallel to x-axis which will intersect $ϕ left parenthesis x right parenthesis$ only at one point.
$not stretchy rightwards double arrow ϕ left parenthesis x right parenthesis$ is one-one.
Foe onto
As, $ϕ left parenthesis x right parenthesis equals open curly brackets table attributes columnspacing 1em end attributes row cell x comma x element of calligraphic Q end cell row cell negative x comma x not an element of Q to the power of straight prime end cell end table close$ , which shows
$y equals x text and end text y equals negative x$ for irrational values $not stretchy rightwards double arrow y not an element of$ real numbers.
$therefore$ Range=Codomain
$not stretchy rightwards double arrow ϕ left parenthesis x right parenthesis$ is onto.
Thus, $f minus g$ is one-one and onto.

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