To find (We fg are(- given two functions3 () f (x),) \ we need to) calculate and ( g(x) ( f (g),(- and3 we)) are \ asked). to Here's find how ( we ( proceedf step -bycirc-step g:

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-###3 Step) ), which means we first evaluate1: Understand ( g the functions -(-3) \ f(x), and) then = x plug the result into \cd \( fot.

1. **Function4 Definitions **),:

• $f(x) = 4x$, and the domain of with \( D f_f \ =) \math isbb{R all real} numbers, \ i).e (domain.,: \( all D_f = \mathbb real{ numbersR}.
• ( g). (x) = x - -1 $$$$, and the g domain of $g$ is(x positive integers, i.e) =., D_g = x - \ mathbb{1Z \}),^+.

with2 (. D _gEvalu =ating \ (math gbb(-{3Z)} ^)+: \ ) - ( Thedomain domain: of positive ( integers g).

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)### \ Step) is2 the: set Evaluate of ( positive g integers(-,3 so) ( \ g) (-3) ) is not defined, because $-3$ is not a positive integer.

Thus, $(f \circ g-)( The- domain3 of)$ does not exist because g \((- g3(x)) \ is) not is defined restricted.

Let me know if to you'd positive like integers further (\ clarification( or \ detailsmath!bb

Here are some related questions{Z}: ^+1. How \ can)).

• Since you adjust the function g(x) \( -3 \not soin that it is defined for \mathbb{Z all} real^ numbers+? ),2 $. g What(- would3) happen$ is ** if you foundundefined** (.

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(g###(x Step)) \3): Con forclude a Since valid input, ( say g ((- x3 =) \5) \ is)? undefined3 due. to Can the you domain find restriction the on composition \( g ((xf) \ \circ), g \()( fgx(-)3) \ for) valid is values also of ** (undefined x** .

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4Would. you How like would further the clar resultifications change on if this (? g

(x###) Related \ Questions): had1 a. different What domain is? the5 domain. of What a are composite the function conditions ( for ( af function \ compositioncirc to g be)( validx?

)Tip :)? Always2 check. the How domain would of each fg function(x involved) in \ a) composition behave before if attempting \( to D evaluate_g! = \mathbb{R}? 3. How does restricting the domain affect the output of a function? 4. Can we define $fg(x)$ if $g(x)$ outputs non-real numbers? 5. What happens to $fg(x)$ if $g(x)$ is a constant function?

Tip:

Always verify that the input satisfies the domain of all functions in a composition before calculating.