Question 128321
a)


*[Tex \LARGE (f+g)(x)=f(x)+g(x)] Start with the given property


*[Tex \LARGE f(x)+g(x)=\left(x^2-6x+9\right)+\left(x-3\right)] Plug in {{{f(x)=x^2-6x+9}}} and {{{g(x)=x-3}}}


*[Tex \LARGE f(x)+g(x)=x^2+\left(-6x+x\right)+\left(9-3\right)] Group like terms



*[Tex \LARGE f(x)+g(x)=x^2-5x+6] Combine like terms




So *[Tex \LARGE (f+g)(x)=x^2-5x+6]

------------------------------------------------------------------------


b)




*[Tex \LARGE (f-g)(x)=f(x)-g(x)] Start with the given property


*[Tex \LARGE f(x)-g(x)=\left(x^2-6x+9\right)-\left(x-3\right)] Plug in {{{f(x)=x^2-6x+9}}} and {{{g(x)=x-3}}}



*[Tex \LARGE f(x)-g(x)=x^2-6x+9-x+3] Distribute the negative



*[Tex \LARGE f(x)-g(x)=x^2+\left(-6x-x\right)+\left(9+3\right)] Group like terms



*[Tex \LARGE f(x)-g(x)=x^2-7x+12] Combine like terms



So *[Tex \LARGE (f-g)(x)=x^2-7x+12]

------------------------------------------------------------------------



c)





*[Tex \LARGE (fg)(x)=f(x)\ast g(x)] Start with the given property


*[Tex \LARGE f(x)\ast g(x)=\left(x^2-6x+9\right)\ast\left(x-3\right)] Plug in {{{f(x)=x^2-6x+9}}} and {{{g(x)=x-3}}}




*[Tex \LARGE g(x)\ast f(x)=\left(x-3\right)\ast\left(x^2-6x+9\right)] Rearrange the terms




*[Tex \LARGE g(x)\ast f(x)=x\left(x^2-6x+9\right)-3\left(x^2-6x+9\right)] Expand the expression. Remember something like {{{(a+b)(c+d+e)}}} expands to {{{a(c+d+e)+b(c+d+e)}}}



*[Tex \LARGE g(x)\ast f(x)=x^3-6x^2+9x-3x^2+18x-27] Distribute




*[Tex \LARGE g(x)\ast f(x)=x^3-9x^2+27x-27] Combine like terms



So *[Tex \LARGE (fg)(x)=x^3-9x^2+27x-27]

------------------------------------------------------------------------




d)




*[Tex \LARGE (\frac{f}{g})(x)=\frac{f(x)}{g(x)}] Start with the given property


*[Tex \LARGE \frac{f(x)}{g(x)}=\frac{x^2-6x+9}{x-3}] Plug in {{{f(x)=x^2-6x+9}}} and {{{g(x)=x-3}}}




*[Tex \LARGE \frac{f(x)}{g(x)}=\frac{\left(x-3\right)\left(x-3\right)}{x-3}] Factor {{{x^2-6x+9}}} to get {{{(x-3)(x-3)}}}




*[Tex \LARGE \frac{f(x)}{g(x)}=x-3] Divide and simplify. Note: {{{x<>3}}}




So *[Tex \LARGE (\frac{f}{g})(x)=x-3]


------------------------------------------------------------------------


e)






*[Tex \LARGE (f o g)(x)=f\left(g(x)\right)] Start with the given property


*[Tex \LARGE f\left(g(x)\right)=(x-3)^2-6(x-3)+9] Plug in {{{g(x)=x-3}}} into each x of {{{f(x)=x^2-6x+9}}}





*[Tex \LARGE f\left(g(x)\right)=x^2-6x+9-6(x-3)+9] Foil




*[Tex \LARGE f\left(g(x)\right)=x^2-6x+9-6x+18+9] Distribute




*[Tex \LARGE f\left(g(x)\right)=x^2-12x+36] Combine like terms




So *[Tex \LARGE (f o g)(x)=x^2-12x+36] 


------------------------------------------------------------------------


f)



*[Tex \LARGE (f-g)(x)=f(x)-g(x)] Start with the given property


*[Tex \LARGE f(x)-g(x)=\left(x^2-6x+9\right)-\left(x-3\right)] Plug in {{{f(x)=x^2-6x+9}}} and {{{g(x)=x-3}}}



*[Tex \LARGE f(5)-g(5)=\left(5^2-6(5)+9\right)-\left(5-3\right)] Plug in {{{x=5}}} 



*[Tex \LARGE f(5)-g(5)=\left(25-30+9\right)-\left(5-3\right)] Multiply



*[Tex \LARGE f(5)-g(5)=4-2] Combine like terms



*[Tex \LARGE f(5)-g(5)=2] Subtract





So *[Tex \LARGE (f-g)(5)=2]

------------------------------------------------------------------------


g)





*[Tex \LARGE (g o f)(x)=g\left(f(x)\right)] Start with the given property


*[Tex \LARGE g\left(f(x)\right)=(x^2-6x+9)-3] Plug in {{{f(x)=x^2-6x+9}}} into each x of {{{g(x)=x-3}}}



*[Tex \LARGE g\left(f(x)\right)=x^2-6x+6] Combine like terms



So *[Tex \LARGE (g o f)(x)=x^2-6x+6]

------------------------------------------------------------------------


h)





*[Tex \LARGE (f o f)(x)=f\left(f(x)\right)] Start with the given property


*[Tex \LARGE f\left(f(x)\right)=(x^2-6x+9)^2-6(x^2-6x+9)+9] Plug in {{{f(x)=x^2-6x+9}}} into each x of {{{f(x)=x^2-6x+9}}}




*[Tex \LARGE f\left(f(-3)\right)=((-3)^2-6(-3)+9)^2-6((-3)^2-6(-3)+9)+9] Now plug in {{{x=-3}}}



*[Tex \LARGE f\left(f(-3)\right)=(9+18+9)^2-6(9+18+9)+9] Multiply and square the given values




*[Tex \LARGE f\left(f(-3)\right)=(36)^2-6(36)+9] Add



*[Tex \LARGE f\left(f(-3)\right)=1296-216+9] Multiply and square the values



*[Tex \LARGE f\left(f(-3)\right)=1089] Combine like terms




So *[Tex \LARGE f\left(f(-3)\right)=1089]