How do you determine $h(x)=f(x)g(x)$ and $k(x)=f(x)/g(x)$ given $f(x)=sqrt(x+5)$ and $g(x)=x+2$ ?

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Answer 1 · 2017-02-03T11:36:56

$(1): h(x)=f(x)g(x)=(x+2)sqrt(x+5), x in D_h=[-5,oo).$

$(2): k(x)=f(x)/g(x)=sqrt(x+5)/(x+2),$

$"where, "x in D_k=[-5,-2)uu(-2,oo).$

We discuss the Soln. in $RR$ .

Let $D_f, D_g, D_h and D_k$ be the respective Domains of the

functions $f, g, h, &, k.$

$"For "f" to be defined, we must have, "(x+5)>=0 rArrx>=-5$ .

$:. D_f=[-5,oo).$

$"Clearly, "D_g=RR.$

Note that, for $h=fg, D_h=D_fnnD_g=[-5,oo),$ &, is defined

by, $h(x)=f(x)g(x)=(x+2)sqrt(x+5), x in D_h.$

Next, for $k=f/g, D_k=D_fnnD_g-{x in D_g : g(x)=0}$

Here, ${x in D_g : g(x)=0}={x in D_g : (x+2)=0}={-2}$

$rArr D_k=[-5,oo)nnRR-{-2}=[-5,-2)uu(-2,oo).$

Thus, for $k=f/g, D_k=[-5,-2)uu(-2,oo),$ and is defined by,

$k(x)=f(x)/g(x)=sqrt(x+5)/(x+2), x in D_k.$

Enjoy Maths.!

How do you determine h(x)=f(x)g(x)h(x)=f(x)g(x) and k(x)=f(x)/g(x)k(x)=f(x)/g(x) given f(x)=sqrt(x+5)f(x)=sqrt(x+5) and g(x)=x+2g(x)=x+2?