Graphing rational expressions of trig/exp by hand?

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In my math class, we are not allowed to use calculators. Thus, when I see problems with an exponent/log or trig either in top or bottom and I have to graph them to solve the problem, I'm not sure where to start. Could someone give pointers as to how I should approach this?

For example: (functions that follow this type of format)

  1. (e^x / sinx)
  2. (e^x / 6x)
  3. (sinx * e^x / cosx)
  4. sqrt(1-e^x)

Similarly, why does sqrt(1+x^2) look similar to the absolute value function when the sqrt function and parabola is combined? Should I be looking at domain/range of outer to inner?