Question

Find the domain and range of the real valued function:

(vii)  \[f\left( x \right) = - \left| x \right|\]

 

Answer 1

f (x) = – | x |, x ∈ R
We know that

\[\left| x \right| = \begin{cases}x, & x \geq 0 \\ - x & x < 0\end{cases}\]

\[\therefore f\left( x \right) = - \left| x \right| = \begin{cases}- x, & x \geq 0 \\ x, & x < 0\end{cases}\]

Since f(x) is defined for x ∈ R, domain of f  = R.
It can be observed that the range of f (x) = – | x | is all real numbers except positive real numbers.
∴ The range of f is (– ∞, 0).