Question 1194818

Given the functions:

{{{f(x) = sqrt(25 - x)  }}}

{{{g(x) = x^2+ 9 }}}

{{{h(x) = sqrt(x -5)}}}


a) Obtain the domains of each given function.

{{{f(x) = sqrt(25 - x)  }}}

since given  {{{sqrt(25 - x)}}}, means {{{25 - x}}}must be zero or greater than zero

{{{25 - x>=0}}}
{{{25 >=x}}}

domain: { {{{x}}} element {{{R}}} : {{{x<=25}}} }

{{{g(x) = x^2+ 9 }}}

domain: {{{R}}} (all real numbers)

{{{h(x) = sqrt(x -5)}}}

since given  {{{sqrt(x - 5)}}}, 

{{{x - 5>=0}}}
{{{x >=5}}}

domain: { {{{x}}} element {{{R}}} : {{{x >=5}}} }



b) Calculate the domain and rule of ℎ ∘ 𝑔 .

ℎ ∘𝑔 ={{{h(g(x))=h(x^2+ 9)= sqrt(x^2+ 9 -5)=sqrt(x^2+ 4)}}}

since {{{x^2}}} is always positive no matter what {{{x}}} value is,

domain: {{{R}}} (all real numbers)



c) Calculate the domain of 𝑔 ∘ 𝑓 .

𝑔 ∘𝑓 ={{{g(f(x))=g(sqrt(25 - x) )=(sqrt(25 - x) )^2+ 9=25-x+9=34-x}}}

domain: {{{R}}} (all real numbers)