Question

Find the domain and range of the following real function: (i) f ( x ) = –| x | (ii)

Answer 1

(i)

The given function is f( x )=−| x |.

It is known that,

| x |={ x   ,  x≥0 −x ,  x<0

So, the given function becomes,

f( x )=−| x |={ −x   ,  x≥0 x     ,  x<0

The given function is defined for x∈R, so, the domain of f( x ) is R.

Since, the range of the given function is all real numbers except positive real numbers, so, the range of f( x ) is ( −∞,0 ].

(ii)

The given function is f( x )= 9− x 2 .

The given function is defined for all real numbers that are less than or equal to 3 and greater than or equal to −3.

So, the domain of f( x ) is,

{ x:−3≤x≤3 } or [ −3,3 ].

As the value of f( x ) lie between 0 and 3, for any value of x such that, −3≤x≤3, so, the range of the given function is,

{ x:0≤x≤3 } or [ 0,3 ].