Question 154421
(FoG)(x)=f(g(x))=3(1-x^2)-5=3-3x^2-5=-3x^2-2



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(GoF)(x)=g(f(x))=1-(3x-5)^2=1-(9x^2-30x+25)=1-9x^2+30x-25=-9x^2+30x-24



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First let's find the slope of the line through the points *[Tex \LARGE \left(7,-3\right)] and *[Tex \LARGE \left(1,3\right)]



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(3--3)/(1-7)}}} Plug in {{{y[2]=3}}}, {{{y[1]=-3}}}, {{{x[2]=1}}}, and {{{x[1]=7}}}



{{{m=(6)/(1-7)}}} Subtract {{{-3}}} from {{{3}}} to get {{{6}}}



{{{m=(6)/(-6)}}} Subtract {{{7}}} from {{{1}}} to get {{{-6}}}



{{{m=-1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(7,-3\right)] and *[Tex \LARGE \left(1,3\right)] is {{{m=-1}}}



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y--3=-1(x-7)}}} Plug in {{{m=-1}}}, {{{x[1]=7}}}, and {{{y[1]=-3}}}



{{{y+3=-1(x-7)}}} Rewrite {{{y--3}}} as {{{y+3}}}



{{{y+3=-1x+-1(-7)}}} Distribute



{{{y+3=-1x+7}}} Multiply



{{{y+3=-x+7}}} Simplify



{{{y+3+x=7}}} Add "x" to both sides.



{{{y+x=7-3}}} Subtract 3 from both sides.



{{{x+y=4}}} Rearrange and combine like terms



So the equation of the line that goes through the points (7,-3) and (1,3) is {{{x+y=4}}}