Explanation
If (x,y) is a point in the graph of a line then its coordinates x and y form a solution to the equation of that line. In slope-intercept form this equation looks like this:
[tex]y=mx+b[/tex]
What we are going to do here is choose two points from the line in the picture and use them and the expression above to construct two equations for m and b.
As you can see (0,1) and (3,0) are part of the line so we have the following two equations:
[tex]\begin{gathered} 1=m\cdot0+b \\ 0=3m+b \end{gathered}[/tex]
From the first equation we get b=1. If we use this value of b in the second equation we obtain the following:
[tex]\begin{gathered} 0=3m+b=3m+1 \\ 0=3m+1 \end{gathered}[/tex]
We can substract 1 from both sides:
[tex]\begin{gathered} 0-1=3m+1-1 \\ -1=3m \end{gathered}[/tex]
Then we divide both sides by 3:
[tex]\begin{gathered} -\frac{1}{3}=\frac{3m}{3} \\ m=-\frac{1}{3} \end{gathered}[/tex]
Then we have this equation for the line in the picture (we take y=f(x)):
[tex]f(x)=-\frac{1}{3}x+1[/tex]Answer
Then the answer is option A.
