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?

1 Answer
Feb 3, 2017

(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).

Explanation:

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.!