Question 198059
Notice how when x increases by 1, f(x) increases by 4 (eg as x goes from 0 to 1, f(x) goes from 6 to 10)



Since this increase is the same for any x value, this means that f(x) is a linear function.



So to find the equation, all we need are two points.



So let's find the equation through the points (0,6) and (1,10)



First let's find the slope of the line through the points *[Tex \LARGE \left(0,6\right)] and *[Tex \LARGE \left(1,10\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,6\right)] and *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(1,10\right)].



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



{{{m=(10-6)/(1-0)}}} Plug in {{{y[2]=10}}}, {{{y[1]=6}}}, {{{x[2]=1}}}, and {{{x[1]=0}}}



{{{m=(4)/(1-0)}}} Subtract {{{6}}} from {{{10}}} to get {{{4}}}



{{{m=(4)/(1)}}} Subtract {{{0}}} from {{{1}}} to get {{{1}}}



{{{m=4}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,6\right)] and *[Tex \LARGE \left(1,10\right)] is {{{m=4}}}



Now let's use the point slope formula:



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



{{{y-6=4(x-0)}}} Plug in {{{m=4}}}, {{{x[1]=0}}}, and {{{y[1]=6}}}



{{{y-6=4x+4(-0)}}} Distribute



{{{y-6=4x+0}}} Multiply



{{{y=4x+0+6}}} Add 6 to both sides. 



{{{y=4x+6}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(0,6\right)] and *[Tex \LARGE \left(1,10\right)] is {{{y=4x+6}}}



This means that the function is {{{f(x)=4x+6}}}



Now if we pick a random x value that we haven't used yet, say x=2, and plug it in, we get:


{{{f(2) = 4(2)+6=8+6=14}}}



So when {{{x=2}}}, {{{f(x)=14}}} which supports our answer (try other x values to confirm).