Let $f(x)=x^2-7x+18$ and let $g(f(x))=2x+3$. What is the sum of all possible values of $g(8)$?

well the domain range is 

This is a 2nd degree quadratic polynomial so its graph is a parabola.

Its general form is y=ax2+bx+c where in this case a = 1 indicating that the arms go up, b = 7, c = - 18 indicating the graph has y-intercept at - 18.

The domain is all possible x values that are allowed as inputs and so in this case is all real numbers R .

The range is all possible output y values that are allowed and so since the turning point occurs when the derivative equals zero,
⇒2x+7=0⇒x=−72
The corresponding y value is then g(−72)=−1214

Hence the range {y∈R∣y≥−1214}=[−1214;∞)

I have included the graph underneath for extra clarity.

graph{x^2+7x-18 [-65.77, 65.9, -32.85, 32.9]}

You still have not said the sum

Here's my best effort

g(8) = g(f(x))    must  mean that

f(x)   = 8      so

x^2  + 7x  + 18   = 8

x^2  + 7x  +  10  = 0     factor

(x + 5) (x + 2)   = 0

Setting each factor to 0 and solving for  x  produces   x  = -5  and  x  = -2

So   f(-5)  produces 8    and  f (-2)  produces 8

So

g (8)  =   g ( f(x))  =   g ( f(-5))  =   2(-5)  + 3 =  - 7

And

g(8)  = g (f(x))  =  g (f (-2))  =  2 (-2)  +  5  =  4

And the sum is   -3

Thanks for trying but it is incorrect