Question
If $$f\left( x \right) = x$$ and $$g\left( x \right) = \left| x \right|,$$ then $$\left( {f + g} \right)\left( x \right)$$ is equal to :
A.
$$0$$ for all $$x\, \in \,R$$
B.
$$2x$$ for all $$x\, \in \,R$$
C.
\[\left\{ \begin{array}{l}
2x,\,\,{\rm{for\,\, }}x \ge 0\\
\,0,\,\,\,\,{\rm{for\,\, }}x < 0
\end{array} \right.\]
D.
\[\left\{ \begin{array}{l}
\,\,0,\,\,\,\,\,{\rm{for\,\, }}x \ge 0\\
\,2x,\,\,\,\,{\rm{for\,\, }}x < 0
\end{array} \right.\]
Answer :
\[\left\{ \begin{array}{l}
2x,\,\,{\rm{for\,\, }}x \ge 0\\
\,0,\,\,\,\,{\rm{for\,\, }}x < 0
\end{array} \right.\]
Solution :
Given functions are : $$f\left( x \right) = x$$ and $$g\left( x \right) = \left| x \right|$$
$$\therefore \,\left( {f + g} \right)\left( x \right) = f\left( x \right) + g\left( x \right) = x + \left| x \right|$$
According to definition of modulus function, \[\left( {f + g} \right)\left( x \right) = \left\{ \begin{array}{l}
x + x,\,\,\,x \ge 0\\
x - x,\,\,\,x < 0
\end{array} \right. = \left\{ \begin{array}{l}
2x,\,\,{\rm}x \ge 0\\
\,0,\,\,\,\,{\rm}x < 0
\end{array} \right.\]