Functions, algebra and sequences
Start with the domain before manipulating an expression. A denominator cannot be zero, a real square root requires a nonnegative argument, and a real logarithm requires a positive argument. Squaring or multiplying by an expression can introduce solutions or reverse an inequality, so substitute proposed answers into the original conditions. Composition means applying the inner function first; an inverse reverses an allowed one-to-one input-output relationship.
For a quadratic, factoring, completing the square and the discriminant answer different questions. Completing the square identifies the vertex and the extremum over the permitted interval. The discriminant distinguishes two, one or no real roots. A minimum over an unrestricted real domain need not be the minimum over a restricted interval.
An arithmetic sequence adds a common difference; a geometric sequence multiplies by a common ratio. Use the index carefully: the nth arithmetic term is a1+(n−1)d. Distinguish a term from a partial sum. An infinite geometric sum requires absolute common ratio less than one. Translate a verbal model into an equation and check its units and feasible values before interpreting a calculated root.