Question

Find the domain and range of the following real function: $f(x) = - |x|$

Answer 1

Hint: Domains of the function are the numbers which on substitution in given function gives out defined result and range are those set of values which are given out from given function by substituting defined domains in given function.

Complete step-by-step answer:
$f(x) = - |x|$
As we know that \[|x| = \left\{ {\begin{array}{*{20}{c}}
  {x,}&{x > 0} \\
  { - x,}&{x < 0} \\
  {0,}&{x = 0}
\end{array}} \right.\]
Here, we are given a real function.
So, domain and range should be real numbers
And from above we clearly see that there is no value of ‘x’ which makes the function f(x) undefined.
Hence, we can say that the domain of the function is all real numbers.
Therefore, for given function$f(x) = - |x|$, ${D_f}:\left[ R \right]$(all real numbers)
Also, form above the given function is negative of mode. Hence as we know that mode of ‘x’ is always positive but as there is negative sign outside. Which implies the final result will be negative always.
i.e.
$|x| > 0$ for all value of ‘x’
Then, -|x| will be negative for all values of ‘x’.
Therefore the range of the given function will always be less than zero.
Hence, range of f(x) is given as \[{R_f}:( - \infty ,0]\]
Therefore, from above we see that domain and range of given function $f(x) = - |x|$ are [R] and \[( - \infty ,0]\] respectively.

Note: Domain is an independent set of those values for a given function which on substitution always gives real value of result but set of value of range is depending upon the value of the set of domain as it is necessary that all ranges must have domain.