1. Domain. I determine domain by looking for domain restrctions such as division by zero. 2. Range. Range is still difficult, even with calculus, However, once I know minima, maxima, increase and decrease, I can often figure out the range. Therefore, even though I've listed it here, range is often the very last step. 3. Continuity. For standard functions, I get continuity for free on the domain. For piecewise functions, I have to do some work to investigate continuity at the cross-over points. 4. Intercepts. I evaluate f(0) to calculate the y-intercept, provided that I = 0 is in the domain. For the x-intercepts, I try to solve f(x) = 0. 5. Symmetry. I look for odd (f(-x) = -f(x)) or even (f(-x) = f(I)) symmetry, as well as periodicity (f(r + p) = f(x)). 6. Limits and Asymptotes. I look at limits near undefined points. If these limits are infinite, there are vertical asymptotes. I also look at limits at too. If these limits are finite, there are horizontal asymptotes. 7. First Derivative. I calculate the first derivative and solve f'(x) = 0 to find the critical points. I classify these points to find the minima or maxima. I also get the intervals of increase or decrease (which often let me determine the range, as mentioned before). 8. Second Derivative. I calculate the second derivative and solve f"(x) = 0 to find the inflection points. I also get the intervals of concavity from the sign of the second derivative.