The points on the graph of a function are all of the form
[tex] (x,g(x)) [/tex]
In fact, the graph is composed by all points [tex] (x,y) [/tex] such that y is the image of x, as written above.
So, if [tex] (3,1) [/tex] lies on graph, it means that [tex] g(3)=1 [/tex].
Since we know the expression for [tex] g(x) [/tex], we can plug the values:
[tex] g(3) = a-2(3)^2 = a-18 = 1 [/tex]
We can solve for [tex] a [/tex] by adding 18 to both sides to get [tex] a=19 [/tex].
So, the expression for the function is [tex] g(x) = 19-2x^2 [/tex]
