Question

let f be a function whose domain is the set of all real number.if f(x) =|x|-x, what is the range of f​

Answer 1

Given : f(x) =|x|-x f be a function whose domain is the set of all real number

To Find : what is the range of f​

Solution:

| x | = x  if  x ≥ 0

| x | = - x if   x < 0

f(x) =|x|-x

=> f(x) = x  - x    if x ≥ 0

=> f(x) = 0     if x ≥ 0

f(x) =- x - x   if x  < 0

=> f(x) = -2x   if x  < 0

as x < 0 Hence -2x >  0

f(x) = 0     if x ≥ 0

f(x)  > 0  if   x  < 0

=> Range of f(x)  = [0 , ∞ )  

All non negative real Numbers

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Answer 2

SOLUTION

GIVEN

f be a function whose domain is the set of all real number where

$\sf{}f(x) = |x| - x$

TO DETERMINE

The range of the function

EVALUATION

Here

$\sf{}f(x) = |x| - x$

The domain of the function f is the Set of Real numbers

Now The modules function is defined as below

$|x| = \begin{cases} x & x \geqslant 0 \\ - x &x < 0 \end{cases}$

Now the function is given by

$f(x) = |x| - x$

$f(x) = \begin{cases} x - x & x \geqslant 0 \\ - x - x &x < 0 \end{cases}$

$f(x) = \begin{cases} 0 & x \geqslant 0 \\ -2x &x < 0 \end{cases}$

$\sf{}Hence \: \: f(x) \geqslant 0 \: \: for \: all \: x \in \mathbb{R}$

$\sf{} Hence \: f(x) \in \:[ \: 0 \: , \infty) \: \: for \: all \: x \in \mathbb{R}$

Hence the range of the function f is the Set of non-negative real numbers where

$\sf{}f(x) = |x| - x$

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