From the picture we can see that f(x) is a piecewise function.
I will assume that the domain of f(x) is entirely shown in the picture.
:. Domain of f(x) is [-3,+5] with subdomains [-3,+3] and (+3,+5]
By sight f(x) appears to be a parabola in [-3,+3]
and a straight line in (+3,+5]
I will assume to above to be true.
We can see that f(x) has zeros at x=-3 and x=3
Hence, we can say that a(x-3)(x+3)=0 for some constant a
We can also observe that f(0) = 9/2
:. a(+3)(-3) =9/2 -> a =-1/2
Hence, f(x) = -1/2(x-3)(x+3) for x in [-3,+3]
Then, for x in (3,5] we have f(x) = mx +c (a straight line with slope m)
From the picture: m = 1/(5-3) = 1/2
Finally, we can see that f(x) = 1 at x=+5
:. 1=1/2xx5 +c -> c =-3/2)
f(x) = 1/2x-3/2 for x in (3,5]
Hence, f(x)=1/2(x-3) for x in (3,5]