Question

Let f,g : R → R be defined as f(x) = 2x - |x| and g(x) = 2x + |x|. Find fog

urgent​

Answer 1

Given that,

$f(x)=2x-|x|$  and $g(x)=2x+|x|$

As we know that, the $|x|$ has values of,

$|x|=\left \{ {{-x, x<0} \atop {x, x\geq 0}} \right.$

So, according to this, the function of f(x) becomes,

$f(x) = \left \{ {{2x-x, x\geq 0} \atop {2x+x, x<0}} \right.$

and the function of g(x) becomes,

$g(x)=\left \{ {{2x+x, x\geq 0} \atop {2x-x, x<0}} \right.$

After solving we get

$f(x)=\left \{ {{x, x\geq 0} \atop {3x, x<0}} \right. \\g(x)=\left \{ {{3x, x\geq 0} \atop {x, x<0}} \right.$

Now we have to find fog, we can also write it as, $fog=f(g(x))=?$

here, we have to values so,

$for x\geq 0= (fog)(x) = f(g(x)) = 3x$

$for x< 0= (fog)(x) = f(g(x)) = 3x$

As both the values are same, so the values of $fog=3x$