Question 275011
Since g(0)=5 and g(10)=9, we know that the line goes through the points (0,5) and (10,9)



So let's find the equation of the line that goes through the points *[Tex \LARGE \left(0,5\right)] and *[Tex \LARGE \left(10,9\right)]



First let's find the slope of the line through the points *[Tex \LARGE \left(0,5\right)] and *[Tex \LARGE \left(10,9\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,5\right)]. So this means that {{{x[1]=0}}} and {{{y[1]=5}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(10,9\right)].  So this means that {{{x[2]=10}}} and {{{y[2]=9}}}.



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



{{{m=(9-5)/(10-0)}}} Plug in {{{y[2]=9}}}, {{{y[1]=5}}}, {{{x[2]=10}}}, and {{{x[1]=0}}}



{{{m=(4)/(10-0)}}} Subtract {{{5}}} from {{{9}}} to get {{{4}}}



{{{m=(4)/(10)}}} Subtract {{{0}}} from {{{10}}} to get {{{10}}}



{{{m=2/5}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,5\right)] and *[Tex \LARGE \left(10,9\right)] is {{{m=2/5}}}



Now let's use the point slope formula:



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



{{{y-5=(2/5)(x-0)}}} Plug in {{{m=2/5}}}, {{{x[1]=0}}}, and {{{y[1]=5}}}



{{{y-5=(2/5)x+(2/5)(0)}}} Distribute



{{{y-5=(2/5)x+0}}} Multiply



{{{y=(2/5)x+0+5}}} Add 5 to both sides. 



{{{y=(2/5)x+5}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(0,5\right)] and *[Tex \LARGE \left(10,9\right)] is {{{y=(2/5)x+5}}}



Now simply replace the 'y' with g(x) to get the function {{{g(x)=(2/5)x+5}}}



So the function is {{{g(x)=(2/5)x+5}}} making the answer d)