Functions

• How Do You Find the Slope of a Function?
  Find a function’s slope by differentiating it, then evaluate the derivative at the input you care about. The result is the tangent-line slope.
• How to Find Values of x Outside the Domain of h
  Find excluded x-values by checking denominators, even roots, logarithms, and restrictions in h; the formula for h is needed for a specific answer.
• How to Graph Functions Without a Calculator
  Graph a function by finding its domain, intercepts, symmetry, key points, turning points, and asymptotes, then sketching its overall behavior.
• How to Prove That a Function Is Well Defined
  To prove a function is well defined, check that every input has exactly one output in the codomain; for quotient sets, verify independence from the representative.
• Show That the Identity Function on Any Set Is a Bijection
  For any set A, the identity map sends each element to itself, making it injective and surjective—and therefore bijective, even when A is empty.