Question

(i) Find the domain and range of the function f(x) = x ÷ |x|

(ii) Find the range of the function f(x) = x - 1 ÷ |x + 2|​

Answer 1

Answer:

Answers are given w. r.t the positions of the questions.

Step-by-step explanation:

For f(x) = x / l x l

For domain, f(x) must be a real number. This states x / l x l should be a real number.

= > l x l ≠ 0

Domain of f(x) is { R } - { 0 }

For range,

= > x / l x l

= > x / ( x ) or x / ( - x ) { x ≠ 0 }

= > 1 or - 1

Range of this function is { - 1 , 1 }.

For f(x) = ( x - 1 ) / l x + 2 l

For domain,

f(x) must be a real number. This states ( x - 1 ) / l x + 2 l must be a real number.

= > l x + 2 l ≠ 0

= > x + 2 ≠ 0 or - ( x + 2 ) ≠ 0

= > x ≠ - 2

Domain of the given function is { R } - { - 2 }.

For range,

Let y = ( x - 1 ) / l x + 2 l

= > yx + 2y = x - 1 or - yx - 2y = x - 1

= > 2y + 1 = x - yx or 1 - 2y = x + yx

= > 2y + 1 = x( 1 - y ) or 1 - 2y = x( 1 + y )

= > ( 2y + 1 ) / ( 1 - y ) = x or ( 1 - 2y ) / ( 1 + y ) = x

= > denominator must not 0 for a real number.

= > 1 - y = 0 or 1 + y = 0

= > y = ± 1

Range of this function is { R } - { 1 , - 1 }.