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.!