Read domains and ranges across representations

Try

A fictional machine accepts only four input codes. Its complete lookup table is below. List the accepted inputs and the outputs it actually produces. An output repeated in two rows is still one member of the output set.

Input code	Output
−2	5
0	1
3	5
6	8

Now imagine the same table is merely four measurements from a machine whose full rule is unknown. Does it still tell you every accepted input? The numbers have not changed, but the statement about what the table represents has changed.

Predict

Predict whether two inputs giving output 5 prevent this relation from being a function. They do not: each individual input has exactly one output. If a second row assigned input 3 the output 9 as well as 5, that would violate the function requirement.

For the rule f(x)=4−x² restricted to −1≤x≤2, predict whether checking only the endpoints finds both the smallest and largest outputs. The restriction is part of the function, not an optional note.

Experiment

Make a second table for the restricted quadratic and sketch the corresponding section of its parabola.

x	−1	0	1	2
f(x)	3	4	3	0

The greatest output is 4 at the interior input 0, while the least is 0 at input 2. Label the horizontal projection of the graph from −1 to 2 and its vertical projection from 0 to 4. These projections visualize domain and range respectively. The formula, rather than just four samples, establishes that all intermediate outputs occur.

Understand

The domain contains permitted inputs; the range contains outputs actually attained. A declared codomain is the set in which outputs are intended to live, and it can be larger than the range. For example, the finite machine can have real numbers as its codomain while attaining only 1, 5 and 8.

A real square root requires a nonnegative quantity inside it. Thus g(x)=√(7−x) has domain x≤7. Its range is all nonnegative reals. To prove that second statement, first observe that a square root cannot be negative. Then take any proposed output y≥0 and choose x=7−y². This input is allowed and produces √(y²)=y. A range argument needs both an exclusion argument and an attainment argument.

Combining restrictions means satisfying them simultaneously. For √(7−x)/(x+1), require x≤7 and x≠−1. The zero square root at 7 is allowed because the denominator is 8 there. In interval notation, a square bracket includes an endpoint and a parenthesis excludes it; infinity is not a real endpoint that can be included.

Do not determine a complicated range by reading the formula’s domain rules backward. For a restricted quadratic, locate the vertex and inspect endpoint values. For other functions, algebraic inequalities or inverse reasoning may be more effective. A graph suggests candidates, but its viewing window may conceal outputs.

Apply

The practice distinguishes complete finite relations from sampled tables and asks for an explicit attainment argument. When claiming a range, state why no outside value occurs and why every inside value does occur. When a table is incomplete, supply a counterexample rule instead of asserting that the simplest pattern must be true.

Continue

Next, transform a graph by changing inputs and outputs separately. Carry its domain with every transformation. A change of formula should never silently restore forbidden inputs or discard a contextual restriction.