(F*g)(x) (G*f)(x)For calculus we are doing review and these have stumped me, I understand how to do fog and gof and such but I'm not sure what to do with equations like (fog)(x) I keep getting...

(F*g)(x) (G*f)(x)

For calculus we are doing review and these have stumped me, I understand how to do fog and gof and such but I'm not sure what to do with equations like (fog)(x) I keep getting extremely bizarre answers that are wrong(I checked). If someone could help me with one or two I can do the rest. Thank you so much!

f(x) = x + (1/x)
g(x) = (x + 14)/(x + 2)


(f*g)(x)=

domain =

(g*f)(x)=

domain= 

(f *f)(x)=

domain=

(g*g)(x)=

domain=

Asked on by komaikai

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

Posted on

With more than one question you need to make separate posts.

For the function `(f circ g)(x)`

this is the same as writing 

`f(g(x))`      substitute the definition of g into f

`=f({x+14}/{x+2})`

`={x+14}/{x+2}+1/{{x+14}/{x+2}}`     now simplify

`={x+14}/{x+2}+{x+2}/{x+14}`    get common denominators

`={(x+14)^2+(x+2)^2}/{(x+2)(x+14)}`   simplify the numerator

`={x^2+28x+196+x^2+4x+4}/{(x+2)(x+14)}`

`={2x^2+32x+200}/{(x+2)(x+14)}`

`={2(x^2+16x+25)}/{(x+2)(x+14)}`

The domain of the function is all real numbers except the zeros of the denominator of the composite function and the zeros of the denominator of the argument.  

That means the domain is `{x in R|x ne -2, -14}` .

The composite function is `(f circ g) (x)={2(x^2+16x+25)}/{(x+2)(x+14)}` .

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