Identify the variable and the given function or data set.
For a function: determine all possible output values as the input varies.
Solve for the range by finding key points (critical points, endpoints) where the output changes direction.
For polynomials/quadratics:
If it opens upward, range is (y ge) the vertex (y)-value.
If it opens downward, range is (y le) the vertex (y)-value.
For absolute value functions: use the vertex and the direction of opening to set the inequality for the minimum/maximum.
For rational functions:
Exclude values of (y) that cannot occur due to vertical/horizontal asymptotes or restrictions.
Identify any horizontal asymptote to help bound the range.
For square root functions: start from the domain restriction and compute the smallest/largest output value.
For logarithmic functions: use the vertical shift and compute the output behavior to find the range.
For piecewise functions: find the range for each piece and combine them (union) while respecting open/closed endpoints.
For relations/data sets:
List all outputs (all (y)-values).
Find the smallest and largest output values.
Compute range as (max – min) if asked for numerical range, or write interval ([min,max]) (or with parentheses if endpoints are not included).
For inequalities of the form (y) in terms of (x):
Rewrite to isolate (y) and use the allowed (x)-values to determine which (y)-values are possible.
Express the final range using interval notation (e.g., ((a,b)), ([a,b)), ((-infty,a])).