Question 318082
Since g(1)=12 and g(-2)=-3, this means that the points (1,12) and (-2,-3) lie on the line. So all we need to do is find the equation of the line that goes through (1,12) and (-2,-3).



First let's find the slope of the line through the points *[Tex \LARGE \left(1,12\right)] and *[Tex \LARGE \left(-2,-3\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(1,12\right)]. So this means that {{{x[1]=1}}} and {{{y[1]=12}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-2,-3\right)].  So this means that {{{x[2]=-2}}} and {{{y[2]=-3}}}.



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



{{{m=(-3-12)/(-2-1)}}} Plug in {{{y[2]=-3}}}, {{{y[1]=12}}}, {{{x[2]=-2}}}, and {{{x[1]=1}}}



{{{m=(-15)/(-2-1)}}} Subtract {{{12}}} from {{{-3}}} to get {{{-15}}}



{{{m=(-15)/(-3)}}} Subtract {{{1}}} from {{{-2}}} to get {{{-3}}}



{{{m=5}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(1,12\right)] and *[Tex \LARGE \left(-2,-3\right)] is {{{m=5}}}



Now let's use the point slope formula:



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



{{{y-12=5(x-1)}}} Plug in {{{m=5}}}, {{{x[1]=1}}}, and {{{y[1]=12}}}



{{{y-12=5x+5(-1)}}} Distribute



{{{y-12=5x-5}}} Multiply



{{{y=5x-5+12}}} Add 12 to both sides. 



{{{y=5x+7}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(1,12\right)] and *[Tex \LARGE \left(-2,-3\right)] is {{{y=5x+7}}}



Now simply replace 'y' with g(x) to get {{{g(x)=5x+7}}}



So the linear function is {{{g(x)=5x+7}}}