f(x)=x what is the equation of the new function shift 8 units left shift 3 units down

Think of the equation of a linear function:  
Recall y = mx + b   for vertical shifts, we just add or subtract from 'b' and that will move the line up or down accordingly.. However, for horizontal shifts, we will need to add or subtract from 'x'.  Note that the slope or 'm' stays the same for each type of shift.  

Now that we know how the shifts occur, we might consider a different form of the equation for a linear function:  y = a(x - h) + k  here the 'a' is just our slope, 'k' is our original y intercept, and 'h' will represent the amount of horizontal shift.  

So to get the desired transformations of a horizontal shift to the left of 8 and a vertical shift of down 3 from our original function y = x, we can make the following changes:   y = (x + 8) - 3  Now you might be confused with how we got the 'x + 8'..   Let's consider values of 'h'.  For positive values of h, the result will be a shift to the right and for negative values of h the result will be a shift to the left.  So since we want a shift to the left we need to use a '-8' and when we substitute that into our new form, y = (x - h) + k you can see the sign change.  

Now we can simplify of course and get the final equation: y = x + 5 or in function form f(x) = x + 5