Q. 64.1( 8 Votes )

A function f : R R is defined by f(x) = x2. Determine

i. range of f

ii. {x: f(x) = 4}

iii. {y: f(y) = –1}

Answer :

Given f : R R and f(x) = x2.


i. range of f


Domain of f = R (set of real numbers)


We know that the square of a real number is always positive or equal to zero.


Hence, the range of f is the set of all non-negative real numbers.


Thus, range of f = R+ {0}


ii. {x: f(x) = 4}


Given f(x) = 4


x2 = 4


x2 – 4 = 0


(x – 2)(x + 2) = 0


x = ±2


Thus, {x: f(x) = 4} = {–2, 2}


iii. {y: f(y) = –1}


Given f(y) = –1


y2 = –1


However, the domain of f is R, and for every real number y, the value of y2 is non-negative.


Hence, there exists no real y for which y2 = –1.


Thus, {y: f(y) = –1} =


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