Question

If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| –$x,\forall x\in \mathrm{R}$. Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).

Answer 1

Given: f(x) = |x| + x
and g(x) = |x| – $x,\forall x\in \mathrm{R}$

$$\begin{gathered} fog=f\left(g\right(x\left)\right)=\left|g\left(x\right)\right|+g\left(x\right) \\ =\left|\left|x\right|-x\right|+\left(\left|x\right|-x\right) \end{gathered}$$

Therefore,
$$\begin{gathered} f\left(g\right(x\left)\right)=\left\{\begin{array}{l}0x\ge 0\\ 4xx<0\end{array}\right. \\ fog=\left\{\begin{array}{l}4xx<0\\ 0x\ge 0\end{array}\right. \end{gathered}$$

$$\begin{gathered} gof=g\left(f\right(x\left)\right)=\left|f\left(x\right)\right|-f\left(x\right) \\ =\left|\left|x\right|+x\right|-\left(\left|x\right|+x\right) \\ g\left(f\right(x\left)\right)=\left\{\begin{array}{l}0x\ge 0\\ 0x<0\end{array}\right. \end{gathered}$$
Therefore, g(f(x)) = gof = 0

Now, fog(−3) =(4)(−3) = −12 (since, fog = 4x for x < 0)

fog(5) = 0 (since, fog = 0 for x $\ge$ 0)

gof(−2) = 0 (since, gof = 0 for x < 0)