Answer :
From the given graph, we have 2 points with coordinates:
[tex]\begin{gathered} (x_1,y_1)=(1,2) \\ \text{and} \\ (x_2,y_2)=(4,1) \end{gathered}[/tex]The equation of a line in slope-intercept form is given by
[tex]y=mx+b[/tex]With the given points, we can find the slope m as follows:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \text{then} \\ m=\frac{1-2}{4-1} \end{gathered}[/tex]which gives
[tex]m=\frac{-1}{3}=-\frac{1}{3}[/tex]Then, our line equation has the form
[tex]y=-\frac{1}{3}x+b[/tex]Now, we can find the y-intercept b by substituting one of the two given points. For instance, if we substitute point (1,2) into the last result, we get
[tex]2=-\frac{1}{3}(1)+b[/tex]which gives
[tex]\begin{gathered} 2=-\frac{1}{3}+b \\ \text{then} \\ 2+\frac{1}{3}=b \\ \frac{7}{3}=b \end{gathered}[/tex]then, the line equations is
[tex]y=-\frac{1}{3}x+\frac{7}{3}[/tex]a) The linear function is
[tex]f(x)=-\frac{1}{3}x+\frac{7}{3}[/tex]b) What is f(6)?
In this case, we have that x=6. Then, by replacing this value into our function, we get
[tex]f(6)=-\frac{1}{3}(6)+\frac{7}{3}[/tex]which gives
[tex]\begin{gathered} f(6)=-\frac{6}{3}+\frac{7}{3} \\ f(6)=\frac{-6+7}{3} \\ f(6)=\frac{1}{3} \end{gathered}[/tex]therefore, the answer for part b is
[tex]\frac{1}{3}[/tex]