Identify the function type and expression (algebraic, rational, radical, piecewise, trig, etc.).
Find the domain:
Exclude all input values that make the expression undefined (e.g., denominators equal to 0, negative values under even roots, invalid log arguments).
Respect any restrictions from radicals, logs, and square roots.
For piecewise functions, take the union of the valid input intervals from each piece.
For rational functions, exclude x-values that make the denominator 0.
Find the range:
Use algebra to determine possible output values.
For polynomial functions, check end behavior and leading term (or analyze critical points if needed).
For rational functions, exclude y-values that correspond to excluded x-values only if necessary, and analyze asymptotes and intercepts.
For square-root functions, note that outputs must satisfy the root’s nonnegativity.
For logarithmic functions, note that outputs can be any real number if the log’s argument can vary over positive values.
For trigonometric functions, use the standard amplitude/period ranges (e.g., sin/cos: [-1, 1]).
For piecewise functions, take the union of the output ranges from each piece.
If needed, solve for y in terms of x or solve inequalities:
Create an inequality that ensures the expression is valid (e.g., root ≥ 0).
Solve for the corresponding x-values, then map to output values.
Express the final domain and range in interval notation (or set-builder form).